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<h1 class="quarto-secondary-nav-title"><span class="chapter-number">18</span> <span class="chapter-title">Limits</span></h1>
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<a href="../derivatives/more_zeros.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">31</span> <span class="chapter-title">Derivative-free alternatives to Newton’s method</span></a>
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<a href="../derivatives/lhospitals_rule.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">32</span> <span class="chapter-title">L’Hospital’s Rule</span></a>
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<a href="../derivatives/implicit_differentiation.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">33</span> <span class="chapter-title">Implicit Differentiation</span></a>
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<a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-6" aria-expanded="false">Differential vector calculus</a>
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<a href="../differentiable_vector_calculus/vectors.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">53</span> <span class="chapter-title">Vectors and matrices</span></a>
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<a href="../differentiable_vector_calculus/vector_valued_functions.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">54</span> <span class="chapter-title">Vector-valued functions, <span class="math inline">\(f:R \rightarrow R^n\)</span></span></a>
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<a href="../differentiable_vector_calculus/scalar_functions_applications.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">56</span> <span class="chapter-title">Applications with scalar functions</span></a>
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||
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<a href="../differentiable_vector_calculus/vector_fields.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">57</span> <span class="chapter-title">Functions <span class="math inline">\(R^n \rightarrow R^m\)</span></span></a>
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<a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-7" aria-expanded="false">Integral vector calculus</a>
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<a href="../integral_vector_calculus/div_grad_curl.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">61</span> <span class="chapter-title">The Gradient, Divergence, and Curl</span></a>
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<h2 id="toc-title">Table of contents</h2>
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<ul>
|
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<li><a href="#indeterminate-forms" id="toc-indeterminate-forms" class="nav-link active" data-scroll-target="#indeterminate-forms"> <span class="header-section-number">18.0.1</span> Indeterminate forms</a></li>
|
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<li><a href="#graphical-approaches-to-limits" id="toc-graphical-approaches-to-limits" class="nav-link" data-scroll-target="#graphical-approaches-to-limits"> <span class="header-section-number">18.1</span> Graphical approaches to limits</a></li>
|
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<li><a href="#numerical-approaches-to-limits" id="toc-numerical-approaches-to-limits" class="nav-link" data-scroll-target="#numerical-approaches-to-limits"> <span class="header-section-number">18.2</span> Numerical approaches to limits</a>
|
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<ul class="collapse">
|
||
<li><a href="#issues-with-the-numeric-approach" id="toc-issues-with-the-numeric-approach" class="nav-link" data-scroll-target="#issues-with-the-numeric-approach"> <span class="header-section-number">18.2.1</span> Issues with the numeric approach</a></li>
|
||
<li><a href="#richardson-extrapolation" id="toc-richardson-extrapolation" class="nav-link" data-scroll-target="#richardson-extrapolation"> <span class="header-section-number">18.2.2</span> Richardson extrapolation</a></li>
|
||
</ul></li>
|
||
<li><a href="#symbolic-approach-to-limits" id="toc-symbolic-approach-to-limits" class="nav-link" data-scroll-target="#symbolic-approach-to-limits"> <span class="header-section-number">18.3</span> Symbolic approach to limits</a></li>
|
||
<li><a href="#rules-for-limits" id="toc-rules-for-limits" class="nav-link" data-scroll-target="#rules-for-limits"> <span class="header-section-number">18.4</span> Rules for limits</a>
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<ul class="collapse">
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||
<li><a href="#the-squeeze-theorem" id="toc-the-squeeze-theorem" class="nav-link" data-scroll-target="#the-squeeze-theorem"> <span class="header-section-number">18.4.1</span> The <span>squeeze</span> theorem</a></li>
|
||
</ul></li>
|
||
<li><a href="#limits-from-the-definition" id="toc-limits-from-the-definition" class="nav-link" data-scroll-target="#limits-from-the-definition"> <span class="header-section-number">18.5</span> Limits from the definition</a></li>
|
||
<li><a href="#questions" id="toc-questions" class="nav-link" data-scroll-target="#questions"> <span class="header-section-number">18.6</span> Questions</a></li>
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</ul>
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<div class="toc-actions"><div><i class="bi bi-github"></i></div><div class="action-links"><p><a href="https://github.com/jverzani/CalculusWithJuliaNotes.jl/edit/main/quarto/limits/limits.qmd" class="toc-action">Edit this page</a></p><p><a href="https://github.com/jverzani/CalculusWithJuliaNotes.jl/issues/new" class="toc-action">Report an issue</a></p></div></div></nav>
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<div class="quarto-title">
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<h1 class="title d-none d-lg-block"><span class="chapter-number">18</span> <span class="chapter-title">Limits</span></h1>
|
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</div>
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|
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<p>This section uses the following add-on packages:</p>
|
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<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">CalculusWithJulia</span></span>
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<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">Plots</span></span>
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<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">Richardson </span><span class="co"># for extrapolation</span></span>
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<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">SymPy </span><span class="co"># for symbolic limits</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<hr>
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<p>An historic problem in the history of math was to find the area under the graph of <span class="math inline">\(f(x)=x^2\)</span> between <span class="math inline">\([0,1]\)</span>.</p>
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<p>There wasn’t a ready-made formula for the area of this shape, as was known for a triangle or a square. However, <a href="http://en.wikipedia.org/wiki/The_Quadrature_of_the_Parabola">Archimedes</a> found a method to compute areas enclosed by a parabola and line segments that cross the parabola.</p>
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<div class="d-flex justify-content-center"> <figure class="figure"> <img 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KNbFemJdLTAPXMRJGMhwtue+pPLtgw+nAIApGjf13E+tUDgfBgcCITTfAEAMKy9405z+1ThfsBmIGHkt8gIDW3oXODoveprU0M1btFZwYN9atYQetOGEYAoOG0ZAHB9SSHfNMlDlfL4UzoAmN55iUgSgL0/NcJFCvqz2SLwxgCAHJxmDD9EQeyRCAQaIr8IO1jCDsBoYSbasAhD2IIVbxs8/9VAkXGnIcPsTSO9HR5Pe9v/FPfx+4bc7UB3jMyhAHDwyMpR0nK7cQMN9gYtXnB232AIKrAJ4RAFEeAN4bAMOJAJTcMNjgc3ZACA4YALGOAHUsMMleIKVSN+U0MIVTMGGFB+UcV2AOZ+PAV/7dMMSzByr7ARyKZsp/F5Csc2kJB9TYMBRdA026CBYNCDFzh23pADshcOvCCC4ZAJGFA6zJB1UuMNyDCFVFiFU8gMH3h6UYMMpUA1vRAB4Md+kKeCOcWC3qN7OfA4GsFRBfBR+meDmhNyEcMBLxCBzRAOKAAJULMKncULrGAIfmB23OABExgOqxAB2hB+Wgg1kBBH0P/wAvk2dCjoXmTYUmYIS4sQAMo3J/uABe4GEvt3Ev1HN3LAC75wir6gAxjwNMnXC1CjBgHAC8CADIT3CgAQieHQBkhAeGwDgrC1CFOzDVHwCptDdJJViSp1idXTDE8Agx2hCgVgcKABh3VzCxH3NHmDPZuAAQz4NFygAlIjCQBwDE5DA4AQDs8QNcsACezYju7Ijtc4Nb74NLcwdv6GBrzgNGF4grTnY7SnjNGDC+mGA2q4EeUgWqQFitQ4N9xABrwYDhhlDE4DBkd4dkLoNLbwDZsQATgHe1qnh20zj05DCA/ZNIHgik2jDes3e+3XjwDZOt+QiQCgBvLQEfilXyL/EYomMYptww1joHpPwwcAcAtOQwIg+TSIh0zfAAoRdwwRAG7W4EjLwAtq1zZjwAElmXpSYwg94AVe+QQ0AIEsOYYuGWRC5AxRwD8x6BFIRgV494bDJ1bF1zWRUAQ9gAOr4DS9AAY4AARIIAffUA0RIJFRswpgAAqWcAgoGQ6jMAabcAjWAAZ2QJJrUw1oEAVAAARRgAb22DSjYII7pwKiOZoqsAti2HHIeFIviToCCQAvwCQesXkjlpMLeTroh3pS8w2JmJKho3MwOYnrlZoktZqj8w2SEAEAgAY16RG/10S0GZenNZeLxIXPY4yAJZwVRZyhg5ZqGRL3l38koZMl/8GTydSI1Qmc2oWdEqWdndMLKOCasPkRA0WDIyGeJEGeuBST0GOdfaWeD8We9nOcySkPwYcR2eaG4VmbyXQMi8k6/KlX/slQAFo3z5CWEbAKBYoR7UZTJmGfI4GfvtaPEdpPEzo3vQA7L8ANIwGN0lgSHioSILpq6IlhZclCxomcZECgInFyGkAOKPGiIRGjzzajKlajKFQNWcA/GDoS9PADUJcSQAoSQnpiD5pWI6pPJbo2JwoAKqCiJKFBdwelCrpvVXpWV3pPWQpUAjoGxDISvvRlYgqd/iWd2lWmZHWm9JSmlsUF/JNEJeEO1mRqcYpjhPaPRCplRvpAvgA7Kv+gDSbRD11QAIXAElH6EVNaYHZqVXhqTnoaNhsJAGx6EtCVVJQ6poYmoomKQEfkpyZxcqPVEpXqEZc6YJk6VZs6Tp2qNa/DpdeAEvRAY1FXqnJKYXR6XbUKVbcKTrmKNZ/KBQFVEmOkBRmaoMMqZjdIpYd6ZqlKP6uKMCYhDA0ATC8Rqx0xqwB2rD+VrN20rFSzqypAOCkBqG0IE+TKEeZqYdmKZ9vaPnf0rCXRbokQE/W6EfcKZfm6afvKSXmDRN4qqtHYWPRqqiHakv9oltbjRWDEEuBAXuYlsBIro6hasfDTryzxqzU2rScxsBpRsHV6sK2WsLMEBo/UsCihBwX/IAZv6RIqmxEsa6wuG20wCz2jdEG80hIDdQLxgLIp+7FDGrKQt6wk2xIHqrRLW610dq2Y+rNFF7SrA00AAEkusQ9T8Ik1sbMY0bPHha48pa7OpKdDW0ovQQkFoAQ5G7FWm2pYS6tae4xcazq65K8qoQwVAAKzWbZMi61O23El6rXSBBOj1gDBQLWDahlNRXx5e657e519Czov9VYxsQ9d0ADvhBNmexFo212Z25+byzn/BLgrAY2kSrqHm7WJu3HayVmNGxMbWwHdoBOlaxGnG1+pC6GrWzedC1cwsUQNQAuSuxK/WxHBq2HDa6XFOzdDxVw0UQcNIAbN67yzq7e1/9tvAIm7H/YSuvBLSbsTz0sR0cti02um1cs2bQVTNBFaDSAN3eu9d9tsl4uv4UumFks3y3VYNHEPR0BXPrG+E9G+VPa+dxq/ZgNe5QsTifBgdXsTCiwRDJxmDqypEEw2s1Vbt1UTe8R5Cfy9mPu/dhbAbDPARlYTbCioPZHBEbHBetbBtvrBXdNhEwwTYiupQUHDEGHDiobDyKrDW8NmOeFgLXrC+ytv/WuwKpxpLFw2Q7ZgObFHGlC4PyHED0HEnWbE6YrEV3Nl0eJhOeEOpOacQOHFDgHGrybGa0vGVaPEOQG67kQUbtwQcCxscpxTbLtMw9ZnL4wTmoBUSjUUe//MEH08bX/cUoGcTJ5mxgIHaTdBDRxbFIt8OTn2tI+sUpFsTIvGZvkbr/8TZ5qMwv5LsU9bxVlDyDwBqQ2Aa0axyQrRyOX2yScVyvnpylYTcAPHE6OayKn8xFQXxS07xavmy1QDaj5BDQ2Qckdhy8oDWYaqzM/GzFIja+rWdDzxe6g8zaosxaysuNr8NMBsyXccuuCJFNScMJUrl8jss9hMpefcNC/WC0AxqgaVFO98ELh8dLpMUrxcS5qFbt0MFJjco0vxzwYR0G830BVV0LKEVekMFL9nYw09zslczrbLzCE3ciUHFKDbAO2sFA5dEBA9eRItURT9Siyle0wnFPz/3BQpTRArfXst/VAvHUr8JH9ZVn8/0WUMbdMcTc8eLb4sbHmYNxTucMpPcdMDkdPvt9MM1dOeBEpoeA6l/BKdWAB1ABVSLRBUvYJW3U9YvUl0JH+aaAe27GCx6xRj7Q9lXYZnrU9pjUlbxIzOSBRazMVGbcx7h9RkKbJQo3sEWRSACrlSMdd1bYl3fU95vUgRFJOaSJNF8cOT2thHnbaRTU+TjUj3w9cYsJZFoQgNcAQQGxUe2g58kAWbUBn+0Ay5QwibIBCWkDvLkKGPnYyfbU6hbUjc05qJbRTgSrhU4aFIsAnmQAaSUBDrMAzS7QV2IBAoYAvS7c3wbM2eXM+Y/7oNNzqTy1kUXXVNVWGfR1Ij68ABsi0Q/KAC1WDdiMPH8Ryd8+zZ3k2ry2Chpl0UBky2ya0SuzAGA8EBhTwQyEADt4ICUZAFoCDUAF3fc3rfqJvfAKZ7r5kUYwRhVmGfrCAHA5GiB4EGpzAQw3AO3MAFcbAQxoIsHmAO9RDjMj7jMv6EUXAN1pDjOr7jPN7jPv7jQB7kQj7kRF7kRn7kSD7kzZDkTN7kTv7kUB7lTL7kUl7lVn7lWI7lQAeq7UDjXv7lYB7mYj7mZE7jslABJoAO81DmbN7mbv7mY/4O9gDn7Q0RA17gB+4P74AB800Q7BABdT7VEk6sFC68Fn5d3P8JhkpxkBWAoOetEtugAi+y3uq8CgRuEH8e6GQ96NZ6zUmtY+7pmtugzkMRD8CaFfbZD0VQCuvABosgEIsAV0AwwufQCyjOBW3AyYXa3Z9OqwKqnHUSyzIVaFfhoewgB0gQCfoCCs8gEO1gBxrFDnbwBFxACt/R26r52+MU3FZUoUr6IsG+E4cc18Xe2RXe6+u1pSLu3qQOFFrcsahu7oaO7mkroDmaPOGOE+UduVvh2Jx+tZ5e2AOGpN9eEPluE//Nof0u79J76IC1pSqwDYjS7j2x4Rf86IJdrPhN71S2pm1q8BS/E7TwS8G08Blf6A3P8UXHpwybEAc/E90gWl//5RX+zt2Kq+3gxO0ntKhc6qgK8fIx8dRP+hU1b128LvDE9alg8PEIAfRePVdw0NVMUfQmdvMOj6wLy6oL4fQuIbfk3hVU38lWr/Jn5a696hBczxJvCu9Ez/Due/U5FUPOOhwhXxPlLcNtf/IBj5qblvU0u/V1PxNNCuBhEfa7PvZID1i7igJhJB2BLxP355ZkYfgJt/cbp/MHFLWO3xOKJa6T7/YNDPc87fcVkfYnEfMakA1mQfmWa/n9hvn0g7Ek0Pil//guIfQaXRasL8+uv2+wP7Lhg0cYYfok8dVhfRa7b9+9T2a/7z2RObMZQfwjgdpHQMy6D/ocLPrm9LZF/xv9tr8SwlBD6iD1U5H8E778P9b82fO3HCH9IFEOIGDeaWH+hI7+Pab+1OO1YNv+3++rAPGjgCZ/BQ0eRJhQ4UKGDR0+hBixYD2JFS1edJijnUVrALKEAxlS5EiSJU2eRJlS5UqWLV2+VPntGkyaNW3exJmTpjWdPX3+zGmMBgASzDA+5Ifv6NKIZxp06cdU6lSqEClWxSpVI0ePQL1+BRsWpUyxZc2ezckT7Vq2KUlhABDlXdaDSelSRVXARby7ff1KvPpX8MKtFTt+bJtYMVqyix0/9qoW8mSv29AAAABJ31+7gy0qK6ABnGfSngOX9lxY4mHKrV23bPxa9uyRkv9p31ZpDAfRZYM7o3aoTkODYMCN0z19vK/qiKxxP6ccG/r0xbapT38bd6NvpcoVxuMx0Pt4psnJV2UO0fl19oxntodv1nr819zkYNZM+vf5fWIKiNnnPAErMm9AptJ7aD36FvRJOgYfxGk+CB3TjTfU9iOPkgJ44MtADxsq8MOLEHRIwQlPdMlBFFdUSUIW0SoFLiTYAQ5D74IpAIRyROQxoRB7jIjEhkx8sciRVDQyyXBcVNIr+zBbZLMauyOPHA0qUAZILf35ccuGhGSIyCZfRHJMFpk0U6djduPAGOVsNM4dFwpAxUsgu7RTITAXEjPNE8v0c0I0A62JFQ4AmDH/quPgRG0fLAqAQ9E8P8Rz0oP2VKhPQhkEdNMFB/WUpScBiHI8RksrpIAj6LFUxEpb9QfThDQNFb5Oa20PVFxPQgYIANo871TPaCnABHVgpRTZjLZbratdH7z12el0lVYkQxFdR0BhBZOmggqoUdbDV1uVFSFaq30uWnRvo7baUQORMlgqgXPHhAJoCVfcfAljtjln121PXYBla/fZZXzlYBgPt+2Lnh8a0GNfA8e1tNyDzh1YNoEzbq1gXF05FIlsF563tDkKmCJAiQWkeFKLDcKY49Y2lhkyjz31Zg/M4BWRYbrycsGdlQdsOc+XC4q5ZshoVro6pQ/+VWEefcYK/zTRhiYaa4OO9ifpphdj+uu2bg4UZACKGHnqkgVTB4QCdNGa5bhj7Ve9f8XW+D28ZyPbzJwxs2NtkksDrwBK5j6vaDu59nrvtcJ2XL6MoU5Y0h6plqof/7pQGXHvFPeS8bsjnwxy0sPqW0mz0bYTc6Y05NByz40DfUvRETu9dL1zt3ldbwIBXHC1PcNRgx1nH692LW/nXffmJ0v9RcqBsdR1jMoZThnZkQdOeSCZf94x08PvKfoVXzm0h3Natd4iORtApXPuj/O+R/DJT2x8/CN89vfgYW2fRPahBUjJb360m9v99ve43S0QLeaD0DKK8KteKCuAEUnVD1h1wM8lsP9uCRqdA8uiPxHCBIIMsgUJAKC+cF3wIcQyFgeT50Gu4K6EI2zgDVEXKv8BwA72yJcLG6IMb4FLhh2MmwJ1CBYSLnElJ4SPBAGAAVxITIgLadu9jjjDJH6wRCF04k+aGEaUQJE9t1ChDs6xPQAKbynvI8gWkag1JZKxQTm0Y/kC9Q3g+RCIK7siQvYxhQKUwYByRE39eFTHPOZkjI2sjZ+asYQpvkJrgTRIP+qgqg0ikn40NAwYIWmTR44SJGZ8DhoBkANzxA2TBdFEA4LmyTlijZGmpEkpTYlK2nxjEQHwoSJ75kaLCKMBV6OlcoT5oVvi8iW6HCUvZdOMJ1TSc5j/7MZwhJHMWg6tmc6EDR7B+RJpukaVOGjlNYkZEXXMSRXc7ObKvjnOmIiTniwpZ+l+CQA1yAN5V3QYxOAZT4nN854ngSYk8/kYalqTe0LcRxcagIVDDpQ0y/SQQQ9akoQ2cqGLwYUK0XlAISpiQ0Kz6Ce7WMONwqSjefxoW74hiQgAAA3+JOk6G0IsEBwrpSqloxeHJMqWItSeRTVJTNfijCgAIAKWlOEFiVgBafxUmaBslg2RmpKX2lGpZwkpAF7gjS0GMItwsypQbSnUMBF1q0c66ltF8lWxzLSmN5Vj++CY1quuNJRalStH4xrYJT2IqU6Fal51ipB7ENKQfFWr/zfZyie3ErarZKTrV3qBArFyg5bW2+SqIBtZeU42U5UN7GXDmFkx0tSm8mAjB10Xy1mOFoF+zSphxzLYwLK2J89oagRWEduoLrYgxkSmbbuHVX8BVrchUa0TfZuTXqigswOlWjayR1zl+gWjBtLoW6O7xOmS0rVkgIdFfdbOAryzu7cNKkufK9j5JvU61eCCU4ebUoYFNGLvhe9a5VtfkYxXh+WFSXUBoIJtWHVbEZ1oRQEsmO8OKLxbNfANEeySTdR0DOl18DoTsaF4cHfCd6mwgC6M1AyXcMNPzG8EQMEPvgqLp+ow8YmRw1y7OXe+LRbhi1PiC+uqQBujPZUwKv+gASPqOJE8BqGPnwtkBwr5JB0GABfmgmThgWM4wcixk7OS4vOsuKhUXqCVSWINLzh1xsqFkzpM0AD3ivnJuG0ugeGq50i+BhhFvsZ7bfS+Qtg5wJIdMJ/RvD81hyQ7WgYwhuhxhAacQcKG9i6Uvyhl3S4af41ms5tpHOl59acASrgHppeL5x7zGbq8laua/0wUZ+h4P3poAA9QqurSkJk8Zm6pp8ln5exkYcsn/k0sT+BTXvda00PltGVh/dYXh1rGo7Z1d3QRmm40e9Xx/aurQSLs8G14GEMhQTMMbZepZsnbdwZ3bsVN7uchuNjHFnNSvNwAfL0b3gIO97ynvdX/6VoDDJjJBLbtzA90nAB+YfZ3VXw9HmBvlN7N861QiGIUXrsjPIW4dMRR/Oy2Rju1A0cqa7NTBXwburH/CbnId8zqKIs7HBfnHV0tg3CFG3ofThGtzJ1N803bHOe5+6rGSdAbb5uUB+gQ+r8RHXBXH/10Ss2OXN6dFxOgIx9RH3q88yxwmxc2LNxQA37ixetta6AbrwS7QybunYof1OqkW2iFlu7vdu8D7nFnyNyVU/d73j1y+YwRomj0bnK4jRZR+TvgFSL44xCenoZ3nDRHlZ93szeO/oi85BFCeeNYfpyY3xsv9c70d8fDBwWoQ12MK3qMkB44pgcn6vGGymvN/yji96BCAbQgv9DTfiIkp6zJ5ap7sUFxVKXydz/gUAANIqT4xrc9anDvTOZ/7YRr+pWbRK4IWe7aINenffZLs31cdr9pEOw9OyB+YlXk6HjWn73xCYT80ypfvCgvquh5vrV7N+SqKoVAP9FTP9JgP1NyP6VJnWXogfATum5pgG1aiASUvAVMDdOaFdRaPgBsqb4xG5ERumzit4bQQMDjwMFowFF6wJq5mXchwHcrhxOgk5BbwbhrQcF4QUiKQZnxGMqRGpljr0RAivzTv4jowb/4wUYKQo4pmNVJG5ErnDrIsR0Euyb0iyfMoyjMGGr5GwAIhK8TukmDOYjQwqjjwv/l8EBzAcH/KztdIUKwu4dHmYJOcog1FLo2vAsvtCMwHBhQoUKwM7Uj6BA1VMIlfAg/pAtAJCNBBBg0GcPAMcST0bWK4EOZc8SsgMQwksR1YZLpAbxUCZr5A71FZEQQ4b8P9D8ME8GNchGzAYIqFDpNKIBli7mE2ESR60Ss+EQnCkV0mY8essS4qz8QGI2L6MWI+0X0eMOLiUNYnMORIEXAKx5pQMXzU8VVXIhnpIpgXKJhrBbbQJ8VWh/AU4YGqAAMxIhm9LcWNAc7QINhkB1tIIR8JIRRWwc/GINbeAhx1CFylBa1MMZu1DELRKujgMd340B4IIFbaIYXWAbL2YX/KhiGjIyKfHiBVbAGINiFZUk0PSPIZ+EJKaIi0esGt6kzhkRIbxy943gFNCiIWwADhNgFO0CIYXiCgkCGIhBJqlO0WDwoa0ghdBS9G8xBqWhIb+NAPziFghgHFcBJGvCCPfiGgtiERSiId4iAnkMIgbyhksSVb+ADwPkjwDtCqmjKZuNANJiFgmCHCECIcDiGb0CfVgoETigIfAAAEFsIFOAADyBMc6iHw0TMxERMZoiLa7CGx4TMyJTMyaTMyrTMy8TMzNTMzeTMzvTMzGyGzxTN0STN0jTN0/TMY0CCKZoFxXTN14TN2JTN2aTN2rTN28RN2ESH15uDechN2JQH/3jwzd8kzuI0zuNEzuTMzXewh980w6PYg6j0B3NAAYbwg03wh0yIhK78yobAgXMITnkAy/4zOqIEp3NaI9FDQ86pirbkNQ48BZ30B2CoAoZYBEvwh13ggoJwBhwISnmrOvN0wD6yA3jYRpfDQz1kypeESYPgQHYgAWdghyLgBdAjg40YhnFoh2HggAabBxIwhnbIglX4z7ELUAKbJGsCx9k5xERkSwZtUC5RjmWIAiA4Bcgbg41whSjoATBgPW3IAiBIuBJtNbKbL1VipeMDvH7ARPN70RiVihXVimiEmWlkMQENxH2yg3lwUMkzRXc40D2E0QaV0gOhUqSx0jPD0v8waigMSCwlBbsR08W7cE9VK9OlEMsSIssmOad0Oog7HRpcVEa/qFNMA1SMyFMR2lMj8SVg6qfJA7tYGlRCHVOYPNQROdOuSdNgW9MSatM3jclbPKZsEIxCNbRLtYhEdaBFZZGwGqnAEzpVWDJSLdVK9UZUrQhVXSBW/RPXwitWFDliYTLPMFU7w1WJ0NX94VUIOaynsopgDQ134w4oLY9WhMNXvNKNCquxYsKI04VjktZppdajONYgydTGUdPC81WcetZ30wVvcUditdVVLFeISFb8Wdb4aFZQbcQChNcpGVdytVZpxNZ0BafNui6LqFcgEYYvM45iFbOFJdKaM1L/B1zXA5VYESGiBljIC5lXRsxYhrhX8slX6gAu/QpTGVU1ZRiOfnvYj13CkOWXkSSwkoUOBXsBz1oKmRUQadCA9vIOiHUynk2IkQ0fm70NuwKAD4tSTNMuOjMVmNU/og3Lc91Ui+tUvMEvUZsKqlWOlaQT8hBaHfParbHaguXUPFIwBpM4OwPb+BFbqcW+ga1StMVaO8Iypm1bJysHtzkceQnY2qNbNLVbu8taGYyxN8OKsi0NpfxbwA1chR1cTS3cwjtcjiGyBTuymZswdXC4QkhZuhjbE2NcuqHZ+kLaycAySBu5zp2T/xqQ0Z2w0jXa50ldx7A2xW3d92Iv2I1d/7lNv8lF17QVoVlTgUD7i9L1i/eBg10E2MjdP6KDtvIUoXsbDOW9C+Z13ueF3m6V3pKjXkZrs2szje5iL0sbnu71XrEr0hPFn1mL0LCDrPPd3pdV3/UFOAAdSvyxXvlNK/ZqXiCRXQCj3bMNX4wbX929qNFirzqo30UBXgUU3qs13Oc5N6JQt0NLqXKwFwfekgF+rwI+3R+73E3pXw0eKHKwF5DzEhDuLhEWSpIsYT8xOJ4jKG4ChxUOXY+933ZlX4p138hROo67YVrKBrdBwjxxYeWCYf2VYdJZuZYjrWSiBr+tngjewAmu3MubYSXZOQAY0sRJKWoYjs9TYixmQf8tPuCvWb2JsSiWbQAzPuMe7tfvTb41Vpqsk2Iu4iaWDds2omO5U+OKrZnNq0HywN6KaFh+2+GgRWMeHOQglpk2dhVuWmRaeODBCWRYtWPyJOSMSTzf45FEhgglS8EWeuQtjOT95RhDvhNaeleO3Zcltq0mNlFWHhjw2zstIeWG+NYKALNZTmU2XOUnHpj4a+TrRSRa8JZgFuZN5uQfLrpPfpYBtJNeVghiwZJMvpxh7sNirtkupo9jyAEKzBNsRoi80ADtGRpaHi1bbl9cfhZknhR0NohYWudL8mZOBGfUFWf2sGZLsed+SBUQOECscWfIgmcgludQoRzxaxV03of/kwGBbnOlffbFfibhXSlBW6zn+bmHMsjF+9NnaP5GjZ6yf34OGgyXXo6HR+EBZrtok4bUTnZFPPaTOswXUnaHI6A+Jy1pmg5VaZ5eajaTQtyXRPa4U3PRuUlovlroaZZkv9EZMnzOnfYc9uoCBXVqjHZGlO40lX4NnV4Z5SUHHISDVHsor45HsJY2P0HqoSndKi4APeBmZHnqtIrqop5qI6lEts40rYFjRSguoR7q/L1lY1aSa4ybsg2GCoDawjbsLrXpa8VpFjlHFkIcr6UFduzYnJpsyiZq8DXqEznI2aHa+tOAeJUtwHZKtz45I6EcX+AeotUQDdBGxQptOEXs/3hW7BU5Ss1GHpndB1wDAVrV7d3ea9Lu6wc57fkJ2X2YvhMg6bJybbeE7RBcEZSsIg7KWHqQKBeQ6eQO7eW+49KOj6NUoyOS2HggpB8AauvebZUd7fNublsh0LT0bmRRh/CYgqZGpLy2KvP25Ptmj2ZYTQwASDmqV3BwODHg6gC/7vfMbjlkEFVSo2RGlnL1WYi5a8QR8J8i8JtGb+jgI7SkpWONZcLmrwm30wqnxvhIUTflJlzNiwr4bHgK8ZQaccsu8dtAUj9NcTvZh03K5xCb7x4n2MvGjUb1IS6Fp0OlB/9wAXKoMRc3VBjP1uv4VB7fkp7ekPFu8STXcoOdDv9XFfIo1xL2+m8NB3EsP9UyJ9502adHFXEgkQa3gYMIH3PllvO7fY5mCC5+HSgpXWQWt60dtyglr1smbw00V2ge6eyH6y5FL/Q/r+BeWtd3/pB9MKnVJjUyr+wl//HF2Fcm9pCQbgAQaLJKh3NjxXTLnY1tJStUH5B4CD7xRrZXj9hY5+LXUNqbcvPhHhB1EIgjiO9E5/Wh9fXTE+uWOFnhGvbaFhAHfwo+V3ZRr+8Cb2jHQNicJd3zgGO7zrdlJ9tmz71n56rzgq1wH493FY+FM3d333YSN3C0qIYsQFlm9w5cxHFMs3Q1H/VGL3WxWFud5XfjoIfpM+hpJ+/JZnT/wnX0EXItve31OPFpF6juctf23mbo314La0jc8Zxd4+gGh5uCZM+2jp86Jw7nxFjbBntx1FjkOlBrXgv4Gkd37lP3kMhbwMzy0sgL+PnwK2f50hrhlA75kX9tz6AHpwABZSh6o/fzgZf4gtcJ493cpmebjLfyiMv5ZIp4yp14n1jdPQ76v8gGh6OCEhO5sB9yqyd7rLeJ3CX5tO8LXfhZkJt6Ljv6gjJguqcJ47WGb+6LffD3lnz7eS95uR9eQP+KE87ou3j6HAlXmYN7Txr7x890oLB7VaYLYw9zh0+rzEekzadgWQcKYHgBWkvjrMjzAsCCeOh7QWN8At759rtc/8knZqyAd5Ajfaov79x3wKz9/CymCk9nR5eVPNNncOKHwU694HSb26mIhy4oloMWPeffItTfYmfXCd5/fabo7zBfQu5nb+gHQgH94jCu/qVQBrcRAwBv/tsPYfWHQvMcYnplCmYGiAKF9vkraPAgwoQKFzJs6PAhxIgSJxrkh48ixowaN3Ls6LFgvY8iReZoR9EagCzhVrJs6fIlzJgyZ9KsafMmzpw6a367tpMlKQwAqrwbafQoUpEhM+7T00CDrqRSp1J1aLEq1qxaNS7dirXkyZQ/x5Ita/YsWpo9f25DAwAAJH1e59LF2HWiuykFTnSr6/dvw6uABxP2eLewR/+wE1GqTOv4MeTIZ9fqNIYDAIlliDfTPQxR2okCR9z142w6q+DTqlV7Xi1RsUTGkmfTrm2bMs6gAKIUde3baOuGtCoU0EPvN/KPqZMz9xq8eULYEWXbrm79+k7cNbnJeRsXOviJzxPSg1NAA63w6iEuX+/+43j10iFSx27/Pv5w2mdaxqz5PYAIxWeQOj8U4EJfASroT3sLOvjQgODN91B9+Vl4YWT7xVSKUEiw82CAEQajQQNauAMigA2iuKJBEUI3oUMVYjgjjWNp6BJ33snF4nrx7aNJAw1oUhqP6qlYJIguNgdjQzLW+CSUM93I0jGXZYZkjw25Q0UBIAiDpZH/F4HJopLMMcmQk1GqueaU4bDCAQAeEjlmc89R44Jo6tAJ3pF7vldmcmculOaahdZ4Y44ALLKjn8wFN1xxxzXKXJ+ThgcocoIqRKihnVqo4TE5AMCBMZbWSZ55FahiKqVisvoniJomxKmntWK335txsjPnq64dVuCBCfbqW6XD/obpb7IiRKutzdaGW6KLGnvsQcKQaOK0xLqa7al0ybOJHLPwg9A7ruxhyDMGlZIJu844pOxBzDo7L2SULQPEqMNw69tSPwapCK/7claswJwh6xEXfjATBSgIvQKJMbeQgExBKKxyyy3hvGvSYmLR+/FtPrkCJxLrFLxaSFt2+eXJ/6cR3DJhB3NkDgnjnkMCowhxwkfFH0IEr0Hygjy0jdB0B0AgOcOMWD3UnNDAEXouPfC2U5sm80a8gGEQByYrpEYpFaMhhy3jNgR0QUITvTZOy/SQr9Wc7fNJAwXUIWnchb2c91xYa8SKHAa9wI1CwNAwT0GvNGMMEoE4hMILNEh+zjz1WH455pYzs9s11nj+Oeihiz466aWbfjrqqau+Ouutm34KnECckznttdt+O+6567477737/jvw9aCjlway2BM88skrvzzzzd8uDzyVOz899dVbz/w7x/+eD0e3jGGQB+cktIwK6wS8TgRKI4TDOdDLY3aMHrM9/0/e7PGWHP9V8+2XMiAUwEM59reZvQkwK37LCDRoUJB2YMAeCGHGC8SXkHcEwIEMQZs/1Ea/Da7kXqMChgULWJd9KKIAxZEHQUQ4GAKqcCoHxAg/cIALewRiDwUBBTT8UQ0O7GIlJluHM+TBDjZs7Wwci438OKjEmIwMAEUwXwhbqJUCNQAEwfDHC6W4ERZq8ShZpIg5sqACOcCjIIZYRj9u4YU1emET/jhHFl7Qg0PAI2AIwaAGlzg0bwTiLXa4SD+i2MWpjKgASpDaFwc5ES4qEj6xOuJ0kqjHJXqQA8AwSCAbKRV6zKEADQBYizTpFUaKciOJLAweJTnJDb4CTj2QYEEyWcr/kYCDB3uRhoBmiRr96RIppyRMKhuzyg3y0Y9Vk2UvOQKpLpwol8mcCimfGZFfDiaYw9xgJX2hEGRKEy9iOI8q7IjFbiYlmuRsCDUBY81rss0WJADAKxfCzXM6BDQNACBD0ilNc9JTIfr0yzrZCbJiAuCPDJlnPxNCwroZB50JFQk/HxrKBwVUoPNaRhEAgAFcOAShEvWHOo7QpSs65J/JjOhHTUqXilrUVhIDgA5geVBBfnREDTgkRFSqS5RKVKdeYWlLDfWNPhaUpts0aj/pUQdPKiKFEPpoR3j6UJ9uBahBVVMzkKDRW0jEo/Ss5V6UIR6ockSqCaWqVqx61Se9/zSm4pQnUsm5zGZKBK2aNGs/7foVSNJHlWul0VD9GFe4StQdWiiAqjKiV0XilZ6LpYpa/2qhZixBo6/IiFefqQsS8QAcXCHrFnkJ2pw+MizClOyMXpoDc2gks7p0xxmCVAi82WW0GWnsOR87lcii9jrfWEQAACAHmbm2lMIwwYFwyRHdahG35GRuUnjbW9s04wmW7UhxGxmP2EbKMLaFoWi/m8/Sduy008UOLt6JA9Zid7CNFEZoTiBWR4pXIs7tJnSRIt3z1gu4AFCDPD6SXS3GoxAmnEM8RpJfFd5Xmgs2yn75mxZnRAEAEbisSAbcQmXgSb5v/Wx9I9LgZz54JP8RlrBZ0guAF3jDKBouID0MXAA4xOPDIA7xQ0aczBKThK8U8iuK0/INSUQAAGgIsIvdq0Jp4AkEUUkKj/en415GOTE+jp95gzzhCl84KS/O2z0UUTcx0NWXOGZPeM88UQedWMs46QUKVtxiLyt5f92wpZOrUuW8TVmXe+ZIm92sFiIbWR425siXl3YPIBWAmVj5s9X6PEtIayTQgo7JM7i8ikO3d5B3Pg8tOE1fNTNE0qWkdEYsfWmX9EIFKyZcVRJdsH2gwoRUkJoBSR2YNJMa1RhR9ar1Q2gyGBorst4XOAyEHqfmWtcLMbUofU0RYF+6Glmw8Ka1cuxp0UMRxJn/Aq63Iu2TQVuT437NlZsE5GDHpNUAUAGstV3nggmjybRgtrid/Wxeq/ncEaF2kIdc5DGU0SvbfpU74FC3M7gD3/nWd0LK3Uh//yzdaFo3u1liDS5YGBTw28rBLbUPWvjPBSz7C8W5JXFFpnxjps34THzhahVowy8hbxQ5lODJ2cYM4hHn95lbbsSXwxwmmxh4wetyczp1mzhH8GxhhG6slQ9S6gsBeG+t4YWOf5wuSwcTh8Epaq1YvVdU72LZo2PxQWH80sCY+TUI8/UiJdyEDL+azxFydi2m/Y5r31Tb3Xx0AHChN4OZ+4pGTiKTsybvB9m7FPt+EKxfVetcRwzi/0GU8wZUQBH3QJnjKwJ0HEveIJRv6dvfbQ3OZH5BTRcN1EEfegaNPsSlL8jpBaqbLBge8/OmU9hBcG9qzR7yLby9P3I/TMtHwOOnaf171FEGu5fZV7On/fWRr/xJDoMGmGnGaqCvnpGXXBkOl33oja9C7f99VoFH7e57bxrxg4caP+C8JmiLHOT7Sf0iZD/RaZk1gMFbZELXzd/vrUjCBckUBBB48N+e+F8BAWB5uZkxeB8JMANy0B9y0IMmaMBe6MLY4V3x1V59USASZdl06QZRJAcH+ka9cZ4i6B90QCCdSKAAoWAkqaBktUUBHmD4JaCCgINeFMAZhJt62OCY4P/g/uhgX/HgWl0gZmhgc7ygabiDUxTAD8wXgCghmDAh3zjhj0FhULGg/G2gEK7HPaACCJrA8CmIF2IJGOaNGGLZdPkgXKgPGmJJDHYeDcJKCWYfeaVgb/XHlaiHFQ4GOHBJoyEhHF7fHMZNHaobGV6TbvCGeySiX2Bh3WwhmUCiCYrXJF5cJU5SonxHJqYhctyDKviPG57fgsQhkkSi1Ywi25XiEhnif7yHJnpFHxZCghWJLBYJLU6NLQIeLnIQh+iKgvRiVixikHSBOoygewwjjxTj0hyj+yXj/JyiHiKiKp6GOpiHFkoDNQJi+oXid2njsrzf/FSJfzyIM0qFOuj/AXGcwBuOiTWyCDbCDDvGizuuTa7IiTyGY2HEwwdWkSYE457s44r0Y8v8Y9AEJMhEyzfyokECRjyw4VMUQvXRiUOiCESejESmDUXSS6iMSqmsyDx+BD20ImJ5JKuEJIiMZMGUZAadpLMMpNewZEbOxT7owtM0ABw4YqPQ5IPYpMDgZB41S6IkTZG0ZGsJgy01mgP2ClI6iFLuC1PqpKdUkr4giVRiBFXaWjZkS1YuyFZyS1dyo600UcmAyVhKhDSIVAFMgXKhJSgKIkW1Xzu6ZafYz1tApVz+5FR0w2Fp4cmx5V7OXlvSD1juyVw2RDl00oGIIEk2Zug9Jts00RP5/8lkKgQ5KNxlwuK0pKWCrKVe9mUADtT9FJQ6BkhoHgQ1fNNe0MLnZaNmOh5nggxYnuOCzKY/KAMjuoAq5KYx7mbe9Sa9eGZPNspk9oMyKEHduABuhqFy+hxzNgtBGZSpjOU+BENV/kAwmGZEZifEbWetVNIlvUpLBiWeiEZ5/h966pt6dkorwZNMfadhSgQ90EJ8UgEX0mcgOuYg7iB3EpV39oombqT/NJqwHF99Ott9qkkl9UK2WKE7JGQDnAE5sNyE6lqFQok76Se3cKA6FAJxaEAhGKWEFuhmHugTBqaC9ieIQJ902qYGUMJHol2I9pqMjqGhYJRGcZTAZN5/Vv/lCaBCjU3aj/ZbkNrhmrTVfmaojaqDJjzoEejCH07ckwZdlFKimgRWUbXM0kkDHBBHh+bljn0p6YUpKUZJVm3V0hwcPeiCgXSJIvSon7mp7cHpLUKJap0DcEKnKmKpluoCcj6Xn54goCLjoQRCcNkBak5EomVDmnrSGbCpYzWqKD7qNtJIdV1X3GjYneYpCOwpVFVqisTmaI3odbzUevFNdmEpcomGooIWq76Hap4mqP7lhfxWcAHY/mTWPijDGRBHBcABNayjpzora1ZgfowqBmBYsRpVOWjC0xSACWgCn6bUs9oWrNKGis1qAXFTPNACdXqSFgRDl5LVrrpHrxr/y7hmCKERqwhl0j5Igx6QyIFogouKa7i+6q8CJGCaBYVZmLXmqzhoa91owByYI4UOrK4W7EQeLFmoGIu1UDzoApcEiRJwqXZSLLxarElibHYQ2pGJ0L72a3UCbIzCKG+abE6ibE4kbJcVEJZuqwbAgTTEa5bI7HLSbFPuBJy9mgB17MfelMgK5/qR7KoSrVfehMAVWqF+J7/6qwvALCbZ6FlBLbhGKyFKRqZh29VOyrEWQmichx40a0I4LYGmI1+ymV8arGS42+BYzT0oQx08aIm460zNrePN67DUq05ULbGd7ZjsrR48aAVQAS18a9cKbt4RLlZKrc3ChLWZbcvE/0MwlAEInkcXRG5XeW1egW1PYa5juJsKbMPJeG4ZEMd5nEEwMGTpUq7PWe6rGC5PEBrBCYw70MIU1E2XwAHgtpbpdqrQjqzYIqhZbNzlZYs60AIjcqsemJ/i+gPcTiDqTpXqloXMvVvNGQs9KIMi8ADxugAlZEP2HsT25mD3fm3zzihZDB4YJN13ggMqYIHs/o8mfChSvG8Txu/pzq+Q/gTzOZ+puIMwJEJ8nocYqMI0ToUAY+fypuf36kTqqUDcTco9SIMmUGfdVIASaII0mOdIVDAdErDy0m1r3oRuFF6jTG8XhO5e6IEw2G5WqLAksnBuZbBNJDAQsog6BEMdbP9rl5QBLQSssSXvD1+wfQIxTaQeCoAfkuxDN6gCHDzwTWkCNaDwDjsxo0LxxBqwlNZE/PGIOygDJUwBidRNA/xAIeQw64kxfvnwGLuwtM6EEIPIPqiDLiRCnpqQBmCBItCxavBwLeLxHZuxmM5E932fg9CDNKCCGIBAkJiQC8ABLWQDGMudHTsYI4uyI8epTKTxe+xDOehCIdyfCZ3HFFCCMkhuIocyiY3yLZdyoMLEAP6getBDN9BCISiBBlQA8Z4AHKhCN3xyEOIuxOnuTErxSkhhBkLHGqvCHKDvK58HFShCMDAxdChycpKxiEqzGf7GNQ/zg77yCYiBJgjDBN//qC23KTkDqS5DKkvgoQGaBj2QgzCggh64cZAQrwYcwRygwixjiTjrZj1D6T2H6jRjIBVqJDj8sx5QARIPdEEfNDwzs4MstD/iMj3PBTtgwiFUg0KYAyRAAnv5wztwwhkd2jpd4hlSxT64AzUEgyoowhkcwa1ucwO4QBcUgioogzp4NIuA9Hk2NJjOxTyoACjAGUofxDqQwCzMAgmYDD/0QCQMwwus5AXV7cXiISq6WDyoAzUogy6oAiXAwRQ88DYjFg90QSLQglEvqrEodWYy9ZvOxS0U0SugAUJYwiIUxCJYgj8cQxEUxDBEgct1jA5ItEacdTeotSpogh64NQ+s/21cb/MJHMEZUIIqBEM2NByR9EP7JvU8U5lIs/ZcBAIpFEQ4qABCZEF7DkMW+AMoHMICRcAQ455Ym+RbBAEsKEMw6AItqAIqaAIlFEIhwAEcdEEXKMER8IALmAAmA3Umv3IDnMAPYMEZFIImjLY0qANe0+pq9ylf/+lcoMEsFAQ7RABCAMF/MAMQ9AMhuJE/4AMA4G9CqAAQFIGAU848FLiBH3iBO8NbIADxdraDm1AmV4AJ8EASnMEcKMIn6EIxTIM4oAOCfziIh7iIjziJl7iJnziKpziCv0M9qLiLvziMx7iMzziNz8M71DiO57iO7ziPz4M8sHiPB7mQDzmR3/+4i7tqQdjBKhTEzSDEE6ykMTi2JUhCQcADAFykP+AAN6wDO6wDlhuEPghCGji3c1OCJqCCKtCCLgSDMCiDNGQDOZSDO7hDPNAgat/5M+n1Tba2ensFKPhBQRjDEiCEHZxCQZyCHfiDLXyPP1zDCzz2RMwDUn+Xni8lnzvpXFT17ESBLTAIIRTFMuDAO7wDDmggPHjANuQDGjTM0FEE0IKjM+sbNJvKYvECDaDAIowLPxSBz5yCCryAoRcEMuQACuwBr2HQOM1epXPlpZ8azSZ76C07Y663oz50Qrx6eEj7asqtgVq7M0V7emM6t8esHo+Vsoe7s1P7p3r7QWA7eGj/u6+qO7SWe11dH7zTa7NH27MvgvmEHjeweujtQU072zK4wvWpAZKrBrJHgcaEHjKowfX1gM843i4QwvWhAOLQe0Qw/Ow9fMRPfN5V/MVn/IIsfMM7nsfPnsTPnsjPHsY/SBWgQRzMPM3XvM3TPA2Awc3vPM/3vM//PNAHvdAPPdEXvdH3PBf0wNEvPdM3vdM/PdQDvQqQQdRXvdVfPdZDfRQAgRxkvdd/PdiHPQqogdEvrGm8QzOkvdqvPdu3vdu/PdzHvdzPPd3Xvd3fPd7nvd7vPd/3vd//PeAHvuAPPuEXvuEfPuInvuIvPuO3PaFeH+RHvuRPPuVXvuVfPuZn/77mbz7nd77nfz7oh77ojz7pl77pnz7qQxM7iBY/sIPStAPJryrIP147JPxJsQP3iNjqj9aPZ37vH0Q+sMNvq8c69AAQoAAvJIQ2vEARqIC7uDQSHL/BQ9UroECAy98soIAOkMAejAt/kwD4h6VEmQMNFAEKHENCeIAHgD+GrQMQHP8ukBUooEAPRAGSGUQggD8JRADPsANABCAxsJk/gwcRJlS4kGFDhw8hRpQ4keJCeWNQAOC3UBKKHF7qGQSGokiOcxVRplTpTw0pf+c8yEsIBJi/ZTg2LjLkjx0JdiuBBhU6lKhBdh5+QiKUsBo8f/OA7PKHL0JRq1exZj2Yxf+Wv20o8iX08C6hHFAvPTjVupZtW4bnUJC1s4lhPxzLeKJwu5evW3zHvGlUuO2FvH5oTvmr52Gcv1Vj+rbNF4Gsvyw1D8Lc2E+FN38kwhmUwypyadMqX6kxCLOhnVJTI7CrfJp27YfvIoT0BwQvQg/m2PUzqC+2QS68bCdXbhBUIIPQcDBspmIjOxTr4Alfvt32O8EJI0kyeAxJP2NIDMKLMI87UHZVDe5JfJAZkINRjOEDoDYTpPb/+ZJEvKcAYE+hnk7Ch4MiXkBiHQAhZOsbFQ4a45aESMIhh8be28gfP1yKUMSr5DMqAu0SsoMTozhAQoUqZhtRxqG88xAhNWL/McgcEvxxJY6DPDhpRojO4UC7QEZBCJkitgJGngIN4mSpIalMaRFMDMpnPxTTK+IVg/jBxx9+FgGjyjMpuuaFg9SYJSF7DDqFyXXg88cQutDM86HRDGonABsPkgeDB8cMSx877NBTUYdqVGiMrl7iwDE7tEMhtEUDBQBOf9C48KAJD+rhGX8GNQhJTFGNkw8TufRHniVeU2gcHlPFdB0MtMOPoXcCwKceAGRiyc1aF43EEoPG0UuhWbJYqBkdiEW10YQACbGaHPzhxYvhKCO2iGOmImGdfuZxKh8STmKHA/YeHROHaqJVVJsX9PEHl239eScse7I4dqFdmIw3T35o/7DGH3g4IEsewxBaBgXhlhgmXCEFPvOYIjZaRY4+6zUIiV4WWgXfitGcNt+wfInCoE2cU5e9ZYBoddFjVAClikT9KUU1f1jJgRQgkvTHGhJAGcMLmUmWEQwwQEEBGoNQKCgQDuKoGhd/ekGDEz886C1pKnehYRQk/PMHEkD8YQaMTQwhATl/kEEBlCx+/JpKfpZQYxMSzDEogr79MccDMQ16JQ5Q7CBBG7up5IMMANqIxCAV8MoHCDksEVc4P6IA5YVhkFbUnFeO8XCdbw6y5pWCDlrnlWE6ZnxGfYZ5heJmnOJmmN2HCU2eY17ppR3ZqfTmlWW0O6dvfJZ5hRdCdZkkHVDiI8QHmFmCM2gZA9cx+CB4agcmO+pHPIZ3ZgzCXbh6eLHlJzCReeVS8umv3/778c9f//3579///wEYQAEOkIAFNOABEZhABS6QgQ104AMhGEEJTpCCFbTgBTGYQQ1ukIMd9OAHQRhCEY6QhCU04QlRmEIVrpCFLXThC2EYQxnOkIY1tOENcZhDHe6Qhz304Q+BGMTlBAQAIfkEBWQABgAsQQAZAKIC/AEACP8A/9n7R7CgwYMIEypcyLChw4cQI0qc+NDfPYoYM2rcyLEjxoEeQ4ocSbJkv3wlU6pcyfKgPpD0rCGbSbOmzZs4c+rcybOnz59Agwr9SWyo0aNIkypdarQo06dQo0qdiswp1atYs149ViKGOpTYtokbS7as2bNo06pdy7at27dw48ptCw7b3Lt48+rdy/futb6AAwseTFgcOG6FEyterPhbEAMTDPj7F9aA5cuYM2vezLmz58+gQ4seTbq06dOoU6tezbq169ewY8ueTbu27du1xV37AsDBJ8uUt+EeTry48ePIkytfzry58+fQS4tjtgaAADXy/BkIHr279+/gw4v/H0++vPnl4IIJChBggQF6wCufn0+/vv37+PPrp6+bVigMACgg2WXc7WfggQgmqOCCDJo3XSu3kABAApoV2OCFGGao4YYcYpgeLMi8AEAGH1QoX4copqjiiiy2KFt/vFgDBAAHELCZhS7mqOOOPPbI4IPHgGMFACXYeOOJPiap5JJMNnnch9GI4wYAKkDQGY5OZqnlllx2aVl/uHwjTiEAePCABsNwhqWXbLbp5psXTvfKMWOZAgAGajQQh3ZHCgfnn4AGKuh46b3izFi4RBCAJQ18tuagkEYq6aSugSmmOMEA+MoJoD1K6aeghhqqOM3MSVYyEmoimqeiturqq14W/3roWNCIOMcujXaKJKy89uorj+J0c0uYZHFTBABapMNpaKz+6uyz0CZI6ivElAWOFgAAMY8ZozUb7bfghiuerGbNQWU7pXkr7rrstktcsMNeShYjAHBwzjsaVNDtru726++/r01brVmoABDBM/wYkOuq/ALs8MMQdyYOMa80c1aiAPiCmroRd+yxuPDe0s1ZxHAAwCupcfzxyizzOm0waDGDAgCHqKZyyzjnLOnEFaN1DQ4AtNGPzQ3rbPTRgsKLy8hndbMEAFHos9rNSFdt9ZIvpwUOGQD0EA9rVF8t9tgsinNMz2ndAQAK6IJdNNlwx51isLjcIlZamtR7jmthy//t99/5Za3WKwFE0MxrfQOu+OLjmf0KM2v1AmAvsCXO+OWYM0e33WsdYzIqsVme+eik3yaOM6/AvFYzMwsim+ilxy47a45DvtY1OQBAxtChvz3778C3Js43uXC+1jdPAIDEPbPBHvzz0H/5TOrgsLU1ADTEw2fvfkbv/fegmd2K7Wz9AQAJ7NTmPPjsMz588Xez1cmd3di2fvv4yy3O9MFU3xYsEYgAMm5zPx2JY0nY+Eb+/iY+8rHFFwC6BW4K6KKxLEkS3Vvg2N5nvLYgQ0KhGA4Ff3TAz1jwS2UhTTRKAQ7zIAMXnwFHzS7DDVMUghPP0KDR9veKXvivLc8Q0R3/tmc/37XoLJ45oQHQMpo7RIAZ4zmGKeaAgT58BhcwtAwy7jDAY3SBFjrEWQPhwo0ZhYF3EzRi2cySxBKS5TJvDE0zgqEcXwwQNL64xTWq+JlCtNAyXVCgZa7xgj+GsWMcjB9dsIUE+BBnhAuKI2dOKElJiucWvyANHzuDDUlcZhskMKQBkJCMQyJyej6My5RoAA/jQFJBlsyMEiupxEtmcjSb5Ewp7mgAcGCgDPUzwM8EaUqHiQMZj5MLmdBHxDRmMEnW4EMYOMGMQkgiDMQARzT48AUDHCMSjNACYobHiTZkog+w6KUkJhGF+oEDCGuwjCYusQRrZOYWaZhEIe4g/5xrFIIPc7jMMZ4QTAN0oxCMuMQfsgkaTGrSip0pRGbU9oJccIMMWSzmv4bHC1r8JS52wsAzRajGFjGiGa3gwCFaCIsXfOMQz0CFSg2gtgFe4wkBHUsXDBAKYmwjArmwDBKAAI5W8AIcHmgFZgSBhGtYRg5uMEAhrtELAPgpEh4w5B1MYRla7LShtxRNLjXDjFJkphtqwx4xNAowHqYyLokKwDGS80oE3UEckSBBMH0BgGDcQaooEA4xlGqALrh0LMxIA00NcIsAOMMywdDCYvk6V8uQAgOHs0woIuAMRhgAES+4jBa+2ksM0NEAfFAVWB/aGUk4FTPg4IQbVIC90/+ytV3HTKZcMgWAdNK1pCyCnBXIcBk75QKKSIgnZogBgDX4AhaRuEMOoZiGJ1ymGZ6F4hx6QEMPKPYyhwAAMuyJg78a4BseyMRlwEECEqwBFQXNDC44QV/63uEP9aUv5TozVtgiQjNp4Co2nMgBKN4WZN/o6EfjgioAqPa3I/XRNzgQQsvwAQBQ7EYEuIoZSQCAFb9ARny7QeHLtEJjBvXAJUwMAMJa5gsYUOAxMgZZAKz1Mr0oQgQAUARiYoYb0QhykF+BCyEHOb6a6e9lcpFRy7yCtAbgBQcgeuBvicMasHhrXGoFgDk005XAZRFfeUkDJFwGw6L8QwAQA0fL8AL/w5cRxB9bcTBtGsDDlTUoB4hrgEuE0jJ+Boc1vgEOA3eDEwDY72cciksqZ2almZHDgy0TCShX2VnHbEUy7mIsZEltOXVFUCQwEwwABNUyKtBMJzhQwiX6wp6m4MBlksGJy5BBsrTghQHsxI3LhIIDpTSAHJZwmS5I9hLdiIR4L0OCYC86rKFRsgGwMQnN3EGimDGFeS/9K47KYsFxuVa25tGcUB8oCsI0aBE8u0QkfBcz0eAAMsgCixlKrn5oJeYLQviHFkaDBCg+BgrAaBlGoNsAwfAAH2RIUz929d3PHo2Go6oZU/DyMsRQwWMt040oMEOU3H7VlbP8Q7mYSwVt/2OOuQM3PAxMIrVh4PASSeBbOIqDF1/ghCYYQYtW92GahYAGZhhRhkDw0hdd0IQg0nDxaDyBEZHgxCuQUIi1PqMQl+AE1JHcGUZ/hhZhGCoQurCG+M5QM7yIgiRkwYk02DbksMr0pvGibHs9Z+X6qapNZcl1zFwD5Bx/bWa2Afht+NjvgoyGj7+BjdJ4PTVl7Qw4jkELZAAe7qP6Ri++nZeCHQw6eM8PI1CwmVoaiBnRYM0kBI95cI1cy3PBGIrvHmYVIaGbmjE9j9jdetcjQ9N6KdnJuhN6+zCDDE+wQhp8rPsdMbn3VtY85/MiM5p5p/jSarWPUHF56Mcdy7yQl/9fgCa069fe++hnkziSAXy9OA1qnyb++dNPfycNb/Pgvsv1vAYe7Nf//w0ycuHHF2rDNuHhfwCYgAfyIHO3F3nDAeQgHgiogBRoH/f3ClHCF4RjOOMxgRX4geWhG7IwgHwhOYlGHh4Igir4HQwIGJ4DAKCDgvO3gjR4IODQCxgIGKwDAK5THilYg0CIHDAifnqBO7qDRh04g0G4hA7CDK1AJ4CBPMrDPObxg0x4hbNxg7CQgX1xPdnzZQeohFg4hs8xhIJhPuhDH1ZIhmx4GkAyGPODAX2XhBHWhnbYHVAyGAAkQPaxhnf4h6V3DblGhHwBQQAgQX0ohoC4iAHjhFD/KBgfBAAVloh1yIiWeBt5OBhBBABDhB9+eIls2B+5QIh8UUYAcEb58YmgiIVvSBji1kj6oYqruISZSBir1EqxqIizuIuyJIjEUhjLxA5gqIa6yIu8KCePSBghVYnEyIzG+IySFwyGohhxlWf7IYvQCIDisA208IuFIXw1ZyDYmI30Ny3JSBgNNmniWIzkeIfkohhc5mUKMo7t2HpKQ4qB0WlaEH8IQo/1GHICwxjipi0M4o//WGXvuBgnl3LzyI4HCYT3yBh4pTcXYpAPaUoBKZGel0MV6ZAXCYI8YzESmQs7NnsNYpEfmT9KwzSMIXwokyEomZLsIzgSWX1nhyEx/ymT3hOSEjkWPwMAZYCEHemMOgmRwrI0PRksTxM1HJKTRfk7NCmR+/c1TemRT2mPZyOSSVmADKkhTnmVo7M5iiSRDxiBKPKVYHk5p5M6SUkWG5hZZ2mVaXlbtdOWY2GCihaXRDmX6SeWdmk2n8MiaMmXcbOWqmOXO9iDKzKYhLlBxzA+f6kbubM7LcKYjVk1iRSZUrg8LmKZl7lD/FNySemF2tOZcvmZ3zNGkSkOaJg+OeKZqLkymbmacTiHKQKbsXlK1LOa4rCHF2eae5mbYaSaq2mIiLgjuCmcG0U8HRSZDTaJyHmayhmWqCSadrmJndgjyTmd7EKcq2mKqOgj2//Jna7HnGP5l6/oSOIpneTJQNXJm2Rxi0synu2JacjkQLwZXsw0n+xZnxuUYB4Fn2SxjE1Cn/4pcs9AcgKKKIpijUpioAcqKnK3oGMBjlkCoRFKKRwVoBSajlqCoRkaKa9nnasZj0LJn8EZooAzoRQqDvrIjwXanyrqMfc3fRQ6kOS2JSA6o2/CDVgGewu6kF2yozzKJuAwWA3YonW3N11iD87gDKxXpIATDbeACs1wDViapVq6pVzapV76pV1KCgbDDOqpozIqpe4igHbRomNBkjTGJkSKpk0id4fBphNjMi8Jp2cqp+WJf4YxThRqk28Sp3zaIwJ4KXVKoT8ZlIP/uqeFimnsl6SJKqDvx5SNmqKPCjE1mn+TyptT+SeEmqkrcqhn0amryZWAEqqiiiItiBam+peIBoGBoqqrqiEXyIWlCqiR+ZaCQqu1GifXMIL4OBavmpR4OSi++quR5IRJqjW62pYvGIO96qjKCik3mIN08aw9mZiRkqzVuh9m6BbFuhhGSJmQ4q3fih+t+BbjmhibSYXnSq3p6ia1yK7aqhikOYxwgq7zeh7hGm73mhitSSn82q/ksa5y0a5wSD+fUrAGGx71mrABq4cB9JvdKq8PyyT/qn8TGxjGGSoOm7HQgbB4obB98ZyiErIi2xwRW7IdyxfYqa/IirEra0CC/ziKgGGyegGeJzopKluzx0Gye6GzJctIZQoqPwu0xPEhzzAYRHsX8ukqSau0tmEprviyeKGf6AArU0u1s2GOifG0IHUnmDqrNOu1GpKQVzsY1dgrXYu2wsON3lgYYusWFuq2Zwu3kVQq50i3WAsXHuorb6u3qKG2+Pq3bmGivzK4hEsaEdmTdasW3XAsVgCjXJu3jRs4pTIwo4m4a4GjMguymJu59mG4Uum5aiGkz8K4pDtJR8mSbRm5Z7Gk0cK6rStLm+upqHsWG/kttnu7X0IxWhmZsksWbmqSqzu6wAseKymgxXunwwcuv0u6Uam7dyGo4TK9jes4wwufsruoPf/rLNoLtyFzntYbF5VqudAyvl5bvQtat5+6LuyrtNxrp8SKuqjKLvO7sn5pv38KF7FqlvqrvMurHIbpv2TxtLzqLvv7sHWJwPfrFsfaLw08r/0Lwf/LFsjgATD4LxX8rQeMwQn8sjsYCADzwcrqnSKssOUavuKCwqs6myI8wmrxrg4Dw5nKQ/0zw2Yxrvl6wwRcwLnxmPjJwxl8FgP7MDgspzJsxDR8FrUZMUsspTpMokb8qr7ZMVPMoyrsxE9MFh+rxUEsxMJjnl7srGWBsh6zxRnqVlZ8xpMasx/DxgfaxWdsLeNkilqwDytDx+35Phx6x2gsbkVgDyzjx9zpxoL//LmIEbV9PMZkLB33ucjWww1aizOInJsbmn+UXKpiKlI5k8moOaKdzBa30KA6I8qXyaKlrBZ3G8qQHMkSk2A22spo0WDVZjSqPJekbMtpEQ0iIgfwCstlK8vFwcq+3DSUmw9Is8tPuanJrDXYQpDNHMvGvEQ/+sbJrLrVXMzXTBvIHM1mQbtW48wfCc3ijBa9ezXm/JC9nM5mcbxj087/GM7wXKF4Sjb0TI7ofM9lgb3zbM23S6r+XBbgCzf7/Izr134FTRbpGzcJvYv93NCGwTX8B9ECTbgETdFjkb8Y7c3fbBqtytFjEcB/E9GXeK24ytEL7DcovYgby9ETfNIZ/02/jkjSZbHBHaw4L+2OOLjSHF3CjNPToSiIJIjTknmEQ13TGSu0HG3DSw3SIb0ZLcvR4FAG2FOaUT3VtGPUw9rQrRm6+szU6erUJB3FmUPUQVjVJJ3Fo6PWNBjTSB3Gb03WtWrWJK3GpAPXIMjWJC3HpcPXFGi1SG0WeszHsSPYCYjXJE3Ihiw7iv1/TFvYaOHIiW3XaErYlF0Wl/w7kY1+YLvZZkGgwPPZ0Ge6m922wWPamLeN3fjVOP3KpY3ZIRraov3PM6OOnk3bEYramw3MXebCgc3b9fm4t+3QlKu+s8PaB5aRx53A0zwPYl3XUk2+FDMrz00W3Bw9zF1Mzf+b3WRBzt/T3Yfkvs/teRsHPuStQzwJ3sZbkviz3ir5uu5NFi6ZP/LdPub93ADdPvkNPvVb3z4JNIyK38RNmOUr4GPxDU+zBMy8QP8NPfv93PEL4QeelgGu4OLg0RZe3SDcDXVjvuBt0mEU4bMTwhouDi2tQyYeOw+c4jNd4heekhes4TotrYfU4pljmNr83EKtUTqulkSc4mTRwmwV5IrTxBoO1UA+4/VYxUROrFjNSgeG5H4zMZAZ5WMR1lXu5NCo5CmO1l3u4UWqyFremxXLbVbumFl+5oZIcJe25lYjDtEAP2c+Fnp9aU4KpQZ7DcxACrgApoI+6IRe6IZ+6Ij/zgy0JQdHO+YGm1t9S+SHjXlyrjObzMkp7titV+li9J6YruGWHXKczjK5RT6fLuCdvele/oebbNBnHlKNB32jjkjZbBanDt6qLeur3ob2rBtRLtu9N+vGRMuffuvHXX26repkHpvvfBbGLtrAPQfCrea7foW97uoKPrkAULn1J+xpKn3P/uyFDbrdXu0QWettIe5Ivd3p5+3rcu0+I+DiXe7LjuHgHhfqztHorYDubmXoDhf53tDyzO/m/oEL3axvEfD+LHwuloD9jmn37hfZ3d8OX/CLDX6wfTvPfdAV+PC9sn6vgPByofDizOAA4OAg6PEip3mvYA18QfLJXOEd/2/xfYnxfQHzvszhM1/vBznSLy/aJK6CKh8qKi0YOF/KK57yNG+PXj0YR0/JMS70Sw93jJ0XTy/INx6EQy8pRV8YV3/GPw6EWz8och0YX+/E16ADSq31U9/cN02uJM3kYt/2bOXXZm/VU46LSzj2cFL2hHH2M8zlTMj3blL1Rk/RYj74dB9Gdv/3De3WWEj4XeL3igH4CPzmbCj5W2L4Xu/PeR75i48/jV/59wzYZKj5c+qLGU/66TzpbYj6WPP28Gn5Larpdgj7STLZC0r7FBrqpx/6Eq76Lcr7Aprqtw/8wWPbihrNsL6IuK8jvs2bxL+auf6Hz39E2xAv/jv9f/8J7Hd4/aOauwjM/W1ZfeplieCfItG/+60c7dM+hunPIcY9/qWs7dx+ifFvq+IvwuQvkADRBQCQef4MHESYUOFChg0dPoQYUeJEihUtXsSYUePGh/+wbeMYUuRIkiVNnkQ5UhyxV87EvYQZU+ZMmjVt3pR5DedOnj19wpwDQEW7lEWNHkWaVGlGjyCXPoUaVepUi+K63cLV7edWrjd1dgUbdmYkABzOUUWbVu1alE3ZvoUbV+5Ccc1eBRObl+tXvX19ogIQwdlcwoUNQ3V7WPFixhtXvmrmVzJOvpMty8wVAYCvxp09f+74EfRo0oetYtV6WTXMyqsnE+MAoFVp2rX/CSe2nVu30rp3Xf9u/VsvMxQADu1GnnwpbuXNnV8UdwyycNfBqYO9hgNAmX7PvX/XyBz8+O9Wcd3adn21dfU/vy0BsCQfefr1GYq3n792b7ztVbP3bydwyABAh3j0Q5A8/BJkkLHopgvwMgAjtOkOAFAgqkENlVtwQw/fMg89CiUc0SdOyiLnQxVt63BFF6ESxxnfSrRsQhpfeiWACJp5scfPWvQxSJQeZObGGo20qRcMAOhFSCcPA/JJKTMKMT0kJbOxRGQ8AACVKb+EK0owx3woxrvAuXKyLCl0RgUAAiEzTqrElLPO6FopMk019XzpGh0AIKO7OgdVik5CvxTn/5tcROQTSz6/eQIAJO45tFKjDLVUSHGeObPRPdMEpwwAaIAnU1NNwvRUF+/M01NH0/wDABLYMUhVWzdK9VYNE13USldfRbITADDoRldjMcr1WP02faUXNH/91EhYIogAGWWvnShZbMljFVoSjfRlSVq2Jdchbct1jldGvY22xGRIACAUdOdN6Fx6d2PWWXa/LfGZFwC4o9Z70bV34P2QeaXVfduNkBsgANBiH4PnLXhi0BLlhZY1F96rRHC0GGgeiwkWbWTwNoVFX475jVAOAF4o1WRyK5ZZMXEQVnhlhts7RFZ0ap65ZKCRw1hjndej0JRhsRl6W5qbhksca1J+9v9oltXDJYIAjoEa26e7VuvmVpKxuroAYQMAFrCv/XptqTCWZeOyu5K7L+IAyMRtZdvWeympqZ7bbPWi+XcOQfu+lW/EjxKb7MAFp66bIgCwQp/FdVX88pMS7SXux4G7DhyBgDhQc1szN12lqVX+HGnqghoq9dOFlp2txlsXrm6wyDKrdlVR9x26bzrXHfeOfwNMsOBPBX75if7m5Rvjc/8ts82cN7V57MtMZuzpqSveJ7Rn295S7ctXiHPPv6detbuPQ7/S8+M3AHrp2W+/Ru24o//Q+dEXRzJe4Tj85U8y74nPfPpHqP9hj3OvsEYBrxO+mwyoQKVb4KAauDxxXEP/FtGT4AQtw4cLZSiDGqTdCR3DDO+FUISSOREHUqRCFDqFhhoBRy9eEQ0XqoeCNMnRjm5YwyFipIO0AGEPX6iXcDGpiHXaoOnEwcJjKLE9P4zJlgBgiidCMYVddEgOYcFDK/pQL216ExjlFEXEHTGJZVwidv4UKDXGiY16m2IrqgjHK4oFUpKiVB3JdMe1gSMYY+Sjf34YqlHFTJBjIiTU3Hi/RJoRLLGalcAe+aVIDi2Pe6xkH7siLGJt0o5f7KIhERlKRXZFFtSylikHicohTpKVASpeE8cly1nasIufvGWEdPeuePHylL4soirJGMxW+sRfANOkMafUSYMdEReU/2SmKHniMIhJTJqQpOUCp/gKUGazmQICGUG+2ctkBqMl5qTQxlwGs3Wyk4bWxCY8tXmTnpHgZ/UEJzL715ty6vOcNlEaBpgG0ICq0JDvNGg8cZK1rTHUngO9yjUjOqIJoU1tFm3oQO1CjI2WCEB3yxtIQxq/h7qkpBytSTRoAAA5HE6l0wxn7U6j0ZfCdCaSo5wCb8rJnKauNyTtqUlnIrqBqGMKQwUTNW/1mMgkVakygR0woLrS4J3mFqmxqk9hwrtPbJWrOrVLf8J61ZckDwtmPatRWVLVtbLVejeAa1wvt1Ow1tWnaEtAXvXaxrT61Ug6uVsGCiDYqBYVbA+iq/9hTZodABzgA4xtrED1ho1unMdXki1RNIoQAAvwwADRxGyQpConcAhQraA1qQwAMAE44CO1X7KHM5xxjcVdoxelQMY1hDtc4hbXuMdFbnKVu1zmNte5z4VudJN7izAIAAAS+IQ9bktUzUqys7d4BmxHxAxE/AsAAeiAVrfL3b6ZKRjgwIZ4/dMNVkRBM7NNABzi8Y/1stdtRBIHfOV7nV/MITZlyQADDoLa/gpptU9Sl5UEPODVNOMS2jlvFwjxgQY8tcGDNRmz3guTCVNYMt14RRfum4NQsCMOCOHHh0FssW7FpMQmzgsx+MClsghioS5ogIzX6Fh6RXip8cVxWCz/jOEAdMEXQjVAkIV8TE9yasRHTjJXUKxiAAAAB6Ew4ZT9R+Ry1ZgmN84yTnTMYw7cwRkMFrMXu1tkRa3rzEhOs02iwYk/nTcKuAhknLNH5mvlq2o1QXOeXwKOW3D5ZZpoB5wFzUBCH8vMFcSzoley4y63+c2Txlylp1rnz2Ja03t+mJ9xoV1QG+vBDTK0TxI9YEY7WgWaYEer2SbqU90sYVuZNWyPwQd4DasNn9b1ruds6W9krHjB9qs1OJFqABRh1ckO2rJ1FWuuQNuqjA7DkoSCa2yTTNu28nXOfuLtlx5DEMUxNrLLXa5X06doWGS3QaVNbWvTY973qvd4/sY6/7DkO5vgFvetc/3vgQW8PMhoYV4Mfkt3wxsDbUiGTRlOL4ena3jr08vEE2mNUiChy9WGBQY3brCOK2fgh5Z4pg/OizWIGwWRQIekV25uY91uMiLvITMY4abAuCHjO69Zy3WjPiwiWuaVvEbJT16ElCN9aEq3zctVA/TvgYPmNmfEWazeNKzTxudbf7oShU70CJCBGBofe9J5ramPNx0nXH8cN1hh8i73gBUqjzvZ5+4j+wkH72X7Rc27jIKwB15vZf9MACP+m8OvbO1dbjsxvOl4t0G+MUxXT+XZRV++A8Dv8dA554Hm+cUUPvRpf1zixe0BQYhD9ZpjvWlYSMDXG/9P6OaNQBc0f3vT5Z4wD1xme0SfJvpGIQB9/3vqid8148ulgx/MZ+/nVuAD036h05dd9aO2e4/B3lvMuDDmhb958Id/8AzK4Q5ptPwINf++OmCFI9vvO/GH7RpIzL4Aob/2KDA287H9257+QwtgMpIBFI4l67ImezIELB8FnAoxSr75M78b2bIVAzMKjB8LhJH/e6MG3MARWbNOO0AQpB8R9Bsq4hMH9Is9YzIngzIWRB8XTAplahQZFIsO7LIvCzMcbMH3Kw8SDEATpJEULAs3kz4iXB4dLAoG9BQf/Ik967MA+LNAg8ITkkKU4MFfsUIBabT7egFIe8IuxJ4vLIn/I8qFJASVE5QMHSs2T0tDNUxAI0wOKoSWMZwJVIvAP2M1PHwiNhQJVQqvffHDRSvDLlM4QqwjQ3SM/+MpRZRDrhi2Yrs4eYNEMJJEKmEGctIZH9y3qbu2TnykT7yIljqaAUQ4RyQ3VDQlVaQIcdgGWqjElVm+iuuyTbxDWVQhWpQIgpqbwyO50us3YJQmYYQIVizGSySxWyCDhItFZVxGPYy8jILDhZk4XjS2o7PGdWJGhjiq1mG3qEPGqgtHgBpHhXDGzwk2r1O8C8G5X1xHT8TG1tNG40m0yys6cLxHi2rH+hmprsMzdJw6dQxIlWpHqmIf+JJHsBO7hRwqZuQr//xhhkNgO7eDO4oEKWHkD/bRu9I7PXv0SFmiRYecHtnrMg/og4k8Sbj6RK8qtcDxx8xjv5iUyXwEkcJqneZ7PtOLPp1MLUOErNZhSQCgPdsjyu36wip5nN9Tv+FryvWSQvcKHPuDPtSrygbTQQBDPANryRXsyg8TQag8GvTDsOCjyrKUMQvESp3RygLJP7eMMwW8tH0pQBX8PrsUM/EzsoWBwAybQL+ctOoTMZhzFSAEABYbQsO8S55Mirx0FSa0Q8hMNtYLzF+hwQi0QcwsN89LzF9hTCEEzX+DPMpME8t0wtNkOKzbTD3BwkAENNdcOaXjNlBpxEeLNNvcuZZTzf8lJLZOa03fRDqHi80bAURVG0TjPE7JFIncnL/dfETnDLx6SzckycRePDaTtE5Qe7B7u5FS7LJk/E7VWy2UITgBpC5qXLjzvD2pOrsI8caLA0j4jE/orIpmM5oIOUZT9Df8bL9I0jr/YLRphMX3FNAB1c/ngTjeu476xLiOXFDiYyPQ8yGpK0+FrFAKjKICNbyvW7x67FAi3KD5dI2bNDoKLdH9+x8MdY2RTEjAa1EcnB8QvYyIXLzGq1FCPB8UnYybdLuc7NEubB4YnQwZ3UrvLNL8PDcqmZoS9IukpD2YbNJORB3Jg9DhGDrMG9IrtUbFQdK8AMolBdNw5BvX0wv/Kq29Mw3ItsmjLe0KqQyM9XPThXwa5CPT+rovv9O/O71Hmrk+Kd0K7hvLvgRUPNXDOA0LtZxKIk3UNzXC+MvAnmBM/PvTSPXIc7GlQp0DA0RUTY1JbeFDnhhMCbxBUdXJZAlDnmBMx1RVu0yVTt0J1uTEWO1KTCnVmOKEHPDMwsRVvzSUVq2J0vzAYMVMMaFVmrBVJkXWQC2qXYWJ2VQ1LnzWZM0pYo1GRzvD3rxW2+wQN8w+OiTOW/3W01yQUiVPLTzFc3VO/EDEaAy3BHVX+GQOfHqJ7Yw3Z63XpsQNYiTPasOFAO1X/EyMlmpPei3YCm0KW6SFUvgDi5vQhS3RP6Y4hjngt4Gl2Br9h1uIgfsKgTP4hF3Y2B6lhzcIAAE4gAUggAaQspJt0X+gBijYA0WAWTC9B3/oB369WcwMCAAh+QQFZAAGACxCABkAoQL8AQAI/wANCBxIsKDBgwgTKlzIsKHDhxAjSpxIsaLFixgzatzIsaPHjyBDihxJcuC+kihTqlzJsqXLlzBjypxJs+ZCZCNKGLBns6fPn0CDCh1KtKjRlVcMODDA76jTp1CjSp1KtepRaHcCBIBktavXr2DDih17lFgfBwCc8CTLtq3bt3Djuv0GqxALAAfW3ZPLt6/fv4ADi4SmaRIZAxJmCV7MuLHjx3GJHQp1SSsWyJgza97MeSbdQrRecQCgQJ6/zqhTq17NeiFhSb1y7cDburbt27gFSw71qxcZABO89Dudu7jx48idfqN1iNYvX5ECCEiRvLr169hZEo7U6zkqA6SFZf8fT768eYjHGPF+DgsIgAtm/p2fT7/+9eWCYP3aTysN8A/2BSjggKtt191+t2QCgAAPGEAcgRBGKKFfyKi33364iAbAAA5O6OGHIH6Fn34X5oKLexmcUE+ILLbo4lDYdMLdhb/wgssaAFhAwIs89uijSxV2QmNvtJSyYIM/JqnkkhqNOKQvtPgyGodMVmnllQrFOCONUCaTAwAIYCnmmFYGOeRztCTjXwk7kunmmy7eIsgrZ/5CCzGoAOAAknD26SeB2ITCCC913hIMMRgAsIAGfzbq6HkValLnL7j0Yg0NALRQwKOcdlodLnNOmksu3fwWg6eopmpboINO2gsu23D/AgAJEKhq662bRTopkdcQE0EAKVSA67DECpbLIXROCuUz1qgAABcNFCvttG+xmsuuUDIDjhYAMBEtteCG65Wuu9qJjDiSAKDCEOK26y5UuQjSSrl2EgOOLxFEoMa7/PYLlLX03vILONGgAIAjm/qr8MIvMcOIpPRW+g04VgAQBsMYZ4xSLoXMS6+J3YiDCAAvwKHxyShv1E0pjFxLby+3YCNOLvm6ImzKOOcMETORaOILvURaI44zJAAAiwsJ6wwYM6S0go3S1cXrMb1QOiMOOEsA4AbUgvliCjavvMAM18V1Y0rLQKPJjDji/AEADoqR/VcYvQgURhpy38azz2mb/8v2LQFgoAyAeY/Etk1pZCIQH08U3lov8vb9yy32isOMBwDw8tXhnXLuUxGROK6a2Wj3fYsv4IjzTREA3LG5546xLU5RsB8Eu+y1s2TKE9+IjhrPl/zcdy68hCzOHQAAMc7NVeGOmexC4T67Qp5DDz2QYWzje2eQT522jcbDAgAH57wezPSP5Y6S+gZJv1D16ENPDCcToeLMQrTcYlAzhfROzPaZIZ3L+gaza7CNGaPxxet8dQ3MgGNsLWHfQb4BMwjaLn63s9wXekeQbiSDIYUoSDNygQxwEGQOdRvINf6AjGQEI3QAfAzwhNc3KEWDbd3oAQAQARa2IcEAyMAMI/8A8EGWSLAgvQgDJyKhAi0844IDcZ4BZKcFC4KjGazowRwWwrSBcOMQnMgFJ74QRIFwIwrcGEgaOMBGDlwiho7xxSFYIbn9QKkZssMREN7xOlQAwQC4iKL04CfFKQ7SI4NM5CEN6Tz3XY9tvIiECQXZyOsZQGYYdN9CknE/hkBjDakTRzM4AIRJEgR9AkkkKrY4EFb0QRYeYGVCJNFAgdzBgszogSk58Qc4bqYbqEAELuqIpmTILk8cYMfrtpEGUhjgO6ms5OweOT1pXlCRhWyfIxPJyG6mkpGcU+QpwzlNKWJSkNHkyCWsYMzZzQEAtChI7QYJDhqUsSAkkOVBwHH/iIHUsyBPuCc3UFBLX8owEsEjpp2OITtkYCAA9+wK2wJxDFoAYBLpJKc3DRnNaqoPm5q0nTetSU6PTnGk09yoJSs5EJlRMp0b4QMAInE4BWlCnqgEJ9t68QKE5FMhJhrINyJAP4GAQwfRIIgVTGHQx8iRjgo1VCi58SWMvg4ZhRAHMQDAh1RagxFzYIYzMhGJUnRTEJlAxBNlR4w/MMIAzDiGR8AxiTv04huckIQgDPgNSdzhF9zgxCQmEUpxHAOthQiiOF4xCR4ujn7iuMUl/jBJ2T2DEZJAluw6IQhogsMZjKhsZBlxiUhY45qyE9giC+IMTQhtdm57xTi1yUhE/5TBp/o0yCFMaYAvWOx+mXgrQRhxmKYuBpiIEJhCcYE62f3mCfrooTjWYEBnAKC4BtirIIrwh24g7xuqS0P2hrYGQ4aiCEkNhQpyK5BvJOO98I3vezt5EFIEgxUqEMTYQhEFcXACTyrggzU0EQCGiuMQSLBaN74AyFaAAwN1A4cK/hiMUIgDBbCIJi1wYC9wREET/iWGr9IIjk5ggIMGCAUfUkeM8moTd6oN6QXBgQMSdGO2OE5lFySB24Rsg8cEcQbrPFAGsxbkFTgwrmCcMYmEKnRUxhNHKADggXZItxWsMGoEnjA7cMxhuipI4yTeKo4w4CCNcC2vOHARgTJCA/8AqMipAbDIijrb+c6skC1CvswJDHQyGABAxpcngYGxsWIOqSsEBzrZjS4Y4A7i8EWgBSKLxq2YGQDoxTTxpcAphsID3PiDODRBAhOCgwxIwB0JDNwFF3eEbaaIQC5wSmuC6CAUPUaIKSIqEGL0IALXPS1BeIEBJQPGF4VAhUL3YyNusPWhFpSoOLbRhfiSAAcmvMb5cNDLYQOAftaARRnWRuMfTvEWALBaR3IhDjJYIZWtAAAu2N0FBhOEGRFw3TZykYbzsfsP2J6iM7qKC3EwAgWpMwCN353KSQCAGAx9QnElfAjZgSMCS0AFMyZ2kGfI1xSvkG9SE2I5EugZnQb/eAIQelc7FZwcn+wdSD8Lgoo1dIMYrHvB0wayVRQbGy7APIRyFdoLWhiQbdd4AQBw/TqRISISUGciB9BnXf0RBHmY5U3Ce/FtQwL8iBS5MI8PN2lweEBxBJHVHSLRCVJNT8Jknlk8Ff4CUc8OF0uPZhoCkMZmBCDDlstcOTMB7AB09SC+wLMgIoHn/yFEHNYogtVnK2ESpHGlLpezAX56EGYYeSDH0MEkwSGJAAhXIMcAwI1/DhcmT4KGxFwW7sIAACv0o+nICIQguRVlVESgoALpAgk0KavzsQ0IX5YzN4DG61MiAwC/kB0SXmBCQAejIG67oSW3SgzZIWL1XGeo/0AuAQDjs00FSxBIKDBgvFKwv5ziuMYtaO94jeDwC8bHRv3Rdw1h61QcOPB5MIcQkwB82eVYA8EHDCcQvRABrAcXyKZsy4YmeCQ7CoIC8DAW1FUQOFJEBtAGf1QQbvBH6DM2leFsBnANEZBlN1UQ0HALMBiDMgiDs/Z4ncABCed3nSAQk4CDBXEJHMBRBhANaeR7CXcNp3cHQCAOwaA/snJ0gAQANaiEh5MGqVYK1xA2a6NwHHB9HOFl9RcaMIVa4mAFb3QQnGcQpzcQd7CGBvAKFzMQYfOAbdENrCB0E1hMuIMvEUBfYKEJMEQQgZBpA0EDfWAQsIBwAxEMrtOAHP/UB0TEDEzFEWVWagKxBlYwSVqgBQZxDBHgf8xgcwYgC0EoEIUwcgaAAxh1CDeGDBFQRNiAA6fHB5w4hCTgOm+1BlGgPUBUBLyVEALjEHPAB5zACZpQCEggV0LIEH0QcwbAAXJwEDZiELigA6snEHIweQYwCY5Gh2PherCnUHdSWM0CZ2JRChWDA3NnAGZhMFEQWtYAANo4EHPQB8jQCz0zSWmACJIRDDTACn9ggBghDiSwBoxwDHeQBtdIAi34g2RADMFwCYywet1QBJKADNxBEF8gCaWwjpzQBcRwC1Ewiaj3ArHxB3/QBa0gW8FQCMSADKyQBtGmEMHIELICADj/iZMRUFkPcQs6UBC+8Ae+dYuIsHMCgQi/OH5a4AvdgAx90JDBV1TeCBYRmIcIEgyFtS0Wc3th4QzOEA3M4H/bkAzREA3NMDbgEEgIcQyvwJRACQtPcw2wAA2UaFgA4AvMEHIFQSoIkZe9YIBp+QqoKBDdAAvKOBDMAAu4cI0DAQ2vwAsmZCgDsQ238ArNR5O/0BAeJ18QBHYE8Q0o8EQDYQ3J8AxlyQzJYErcYFUH0Qy8xAkz+Q0k4IdTWRXd0Ap4aJUSgzuRoC7x4CKcwAE+RxbBoC0W4ZnDhYAMsWsTYQquVptV4XoHkodQhjsNGAG82CJfEAXj0Q1IYJQLMXMR/wEOSCCa0EkVLSmBVvkq24A7RAMALwcizFAGEaACbjCcydELeMMQnjcRhZBl5zkVt5mb62l0uAMOTwAAZfAgLDJ65PEKTKcQnCCQDHELUBmgTyGdVmlHsvAM0iMIAEAD84ChLoFmCoGfDcGYJFoWiKCeVpkt0oMLgaN5K1qj83GbguAcG8of5+KeRaOWNhqk9PEMlzAJ07mh44g7qwMAziikTmodxNCiO3ohAlNYbCNTOEAPT7ql2PENrZCjU4ohdyU9sgAAGIAOXJqmyQENmWCkYfoLICM9l0OIalqnuSEZpfCmvXELRyc73+AeumengtoaXgoaelp0r4U77wQE+f8wqI6aGmzqpm9aNYP0CuODpo+aqZqBp3qqNoOEDKNRf5o6qozxGTqqp7TQo7jTDTowU6T6qovxGkcappRjpWzjH0hwErC6q32xG52KIM0lPXlSZQzKq8ZKFsvRHL8Kp24nPceQKB54rNIqFgayrM02SNiAKRc6rdxqFb66rAWUSM+lq91arlORrKfaqTaUSLJCAhlorvAKFdW6rHeUSL4SAM0Qr/p6FOmxHvSaJomUdADgTPtasEHhJMvKHwY2SF0AAFrAlQYbsTUxYFuSsFKVSA6HgRK7sTNRIf6asMxlq2zDhxTKsSZbEgibsMwaZbhTMAAAoCcbsyihJbP6q9f/Sk8VQwbFKrM86xFmorL7UXR9Kj0j8wK/2bNI2xHLUQgkArSyl0g0EwEqmrRUWxE0C7R2RAsVOEjvuY5V+7UT8bNY2yWKhDVaA7ZoKxFykixYWy8iO1FvM6JpO7cLwSqE0raTg5WKBDgYQKN0+7fkgre7mUhzqjl/e7gFASps27bVmUhL2qSIC7Z2i7dBCyvYhDyMGrmIG7h4K7TYJD4cgKmaS7fxsrht6wsdik0IdJejS7cAQ7meqkg5BADi2bpg6zAQA7t+g01ukDzRZbtgyzHeQ7lJqkisMD7KBLxV+7q6W6XY5FAFprxVi7u6eyGDm0jdUFXSS7VSU737EafY/1QGAJCr29uzK1M61Ruu2DRlVVa+PLs34Ui5iApSiBIAM+m+G9u93osmVoNN14ApO4i/JitA+xu72ER7tifAHAu/Bby72KQJAIAC7qDAG9s9DTw5lYNNwZAvtEnB8UrADXw6bys75UiSHqyvM3TBjVu23BIGEHvC8Ao5UNXAN4tNvakCRwvD5grCDay+2HSd2anD3ZrCF7yuIOWy8SnE0vpUF8yhW1u2UXBdO6vEu4pcw9TEZAtS4lAIJJPDVGyszCAJTtbEtLCw2CSjffvFS5xsTUylegtS7wmkagyryDV0TRyyWkyeABCNc7yrTDbGTbzC2ASJWdrHsFqVbcxsuP/gbFpcphhQPoY8qnZIoG3suVo8pykUyZkKjomctR6qxX8KAN2myY/KMS7axvWqxeKwqI1KyoMKDaTQB7AwKrRcy7Z8y7icy7qsy61ADNfwy8AczMB8vH27F65sp0wmCQPUycULUqAKfccsqBHoCzV7wRerxawKAG4YzU86oAJDzZ1MKWOqyjiCBL/LzVuqoc9RzfsryNg0rO0wxei8ounJJeycvjGjyuLgUJM2z07qzU9yz7BryVqcrQCAdv4cpOpsz1icuvo8rgkdpFF6yhcCzg0Mo/rcru8a0SSKo+ka0BecqvqsVfmSrxxNokQqqcoi0Fhbq/ossBF60uc50UD/Y9He67z63LAPK9Pn6aVgWtMsDbLjrMrkp7E8XZuRGtR2pNSdCr6qTLJHXZucKjk2Lb98OtLR4CwwG9WsZ6qxx9RhOr/6TDFSzNUPSBgqTdVgvaOUOtJDZLRmzXrfqlBV7bS0sDYjHbVTG9dw5NUTWNcqK9Ij/Z6ywNdKJqsvutZ56NJjnaBtYNjGNdd5CNi/KsIjLQ4gigNyC9kxhK5TStl66s4ghcZ+y9lyM69srdjEVMOq3AxFU4OmvT2Sndoq68Oq/LixvT3J2rRhCtpsTQvaN9KYuxa5XTioPamqnTao+8SqLD5nWtyO47EJ69uTDbCXvbqdBt1kk7K/St0T/1jGl6067hFC2k02FJvcat2p1zzSvZu55Q010j226D0peHzZx5tM8vzeGMPdKuvddSTazvxQh6nfKXO1eOvffWMj7XnZ2QsAQEbgKSO2Bz7fNELQI42r5ArhGcPfp0vhnhze4mAKVGZlGq4xWnK3uovg5YLRl40o/VziGCPhKU7hWXzZ/9t1MI4xinvRFN7MI/0bCZzjCjO5PL7Yb3zZECzBQq4wnFvAKn4m1zvSG9yHS+4vx2K6Tp7cAA5S5SiAVS4uzIvFqg0zmHTZWtkFL/zl4NLkYk5MFj7S6YLDat4uVx7OS11HqPvJ4X2d4Dnn0hLmnfzkLH7ZSOzn4EK9dv9e0WAt2OHtYWVt6NIivInO0GnD2OF9CF0M6cUC6ImO4Dgd3qSt6cSC6JNO6eUS5YNdNPMo6qmiv6Vu6qKymCB+NVnDx6yuKue7zK9+57ti25ftNoV866nCwLsO0q5CC4ka3hb1yMLe6pFT7GcC2m0N4pjc7J7Cw9AO6xV917Mu3qJs7Z1C7Nlu7EPC6CCOua0M7o1iweNeJ4Bt6eFtqaGr7o2C7e2u7fUN4s+cmfTuJ0R879F+pFuOzTq0zf0+Jr3wnwCPLdPJ2uFdzgKRNAePJWYjTAvP8EF71d2OTOygBN8y8Vjy7xcf8GgS3Pou4CBPJkw88gy/3N0uDtyAAwD/cIYpfyVWzPIrzgvW3e3PJR41byV/HL84zyWvYMYg3gmzogw/byWIPPS7Qgu9MMKqXL+6sPRMEnR27PRQ/pcvL7BSafU/EvRavyujwg2MPOsIDPZK0vRjfyYKDg5nH95FrQfCYhpq7yKTnPVtfyGeC/ezzoeAcDP5ffcQwsl7H+208Ml+H95ZDQB4wCiE7yJsf/jbjtdXE/eqrJVSEPl4f4d6T/l20k5ss/hurS5DIPGcPyGGD/o04uOkr8rXmQKQn/oeMvmsn7e2+vpwXDRQ8PG0HyEAffs0guqX39gAIAO/7yELLfwrm0i6r0gg+gJtkvwQUs/Mz/f57PyYn0ho/8wAzEP9ARL8188r2PT80uPapDH74G8fy8/8qNu/Zbv9Sso6F6D+608fND3+HGr58Q9SAMEHwAYCWwz4M5BQ4UKGDR0+hBhR4kSKFS1exJhR40aOHT1+BBlS5EiSJU2e3PitlSBav1y+hBlT5kyaNW3exGmTFjJxPX3+7AmOG9CfsABgeIBS6VKmTZ0+hRpV6lSqValC0zSpV06uXb1+zXmLGDiiRIWWFceMA4ABH6y+hRtX7ly6de3erUrsUCmwff3+vXnrF1m0P88S/QYEQIYGeB0/hhxZ8mTKlSGqZAlY82a/uHp9Kwz0MFA5ACwQsJxa9WrWrV2/NgAtk1bOtf9t38yFq1to0UOBsgIg4EEq2MWNH0eeXPlFvXxvP4fusteta7x7A0W2dsBy7t29fwdf9xusQi2jn6/di5Y169d9dsMBAIGZ8PXt38ef/yJW2uj9A/aFFmfac6+nMkxLoh79FmSwQQeP0yuU/yb0K0BmCCwQFQAcGOJBDz8EMcS5xiuPQhO/2glD94jBAIAFRIQxRhlnDAkrSbY6MUecxCJMxaCGwkYE+cqhsUgjj0QyQh2XtAkXX3r0UZyziACAgoOQxDJLLRf8hpZDzGMyTJhyyWW3KH0SKpEAHIiFny3fhDNO5LCKBEcx7+QFF9/ODMoTDALww005ByW00MiUvDP/0emq49OnZiQAwAlDJ6W00qq+keUQWBJNNMBoGvXpGgoC6KAfS09FNdWR6LSTUyZ9kaUZUHu6ZgQAJnBHVV135XUiZBiR0FUxA0xmVnGsKeFWaXplttlduxRkU2HFpOUYY50RcoJsnOW220mt0aTOacW8JRgoz2xGSAuM8bZdd7f8NdhxmXTy3CiTCaFKdt/lt98YoZV23iXJNLNRYkgA4IBZBPW3YYcXxKYTcQVeMs9tZvVlrQsqQehhjz8G79dOKGZSPUYbzaXFDOLoGGSXXzbulmhJXjLAZ2aFJQL5YOa5Z9awCYURXmjW0cJZUdE5AZ+XZjqyXzUheklaigWV/xMAAFCgaa23ngsXQV6JWkdaxgKVEQACyJprtdeGCmihw86xXHtVNDuCBSqohm299y7pabhz9Aw0PsGZ46i7+UY88Y5yOQTsv00kuFFwDuSAAcUvx5wit3N53MTpsGm0Gy0ACOEBDTJHPXWF/O6cQpMb3eYJAFR4wC3Vb788l69bpxDWAfm0RjEchqgAd+P53pz3CWG9kE9ncgBghw6Pp35tZhiBWvkJU+STmRcAQCIJ1Konv+lcCmlF+wl55DMZFQDQwovxy6cf5uTV90+wuds7GAAy4Jhf/QTosetpwhf480/g+JQxAMxhgA/8mO7Sh0D05KZgPsJFiw5RicZA0P+D7+pGKRjBOQqeZ1F8yhkANPFBFrqLGZEwYAlNuB4+IQ0ArGhhDrklQRmeJ0C/i5LVAAALHRaRV90wxQh7GB2jnaluvTBiFFP1wksccInQ4V6U6oYMKXaxUr0QxASv+Bz2+YhwR3GGF9VIqGckkYRjvE29ojQ5AHigG2vEI5yOEYhI4IJMfwRkIAU5SEIW0pCHRGQiFQlIWtwiGteAZCQlOclIQmN0JLhjHjWJJV8cwhRwhE6e9oQhbMjuBezYZCqL1A1UIMKPoLzNCX0UPADkoB2qxGWMqHjAN8JSM56K0vMAUIR45NKYIOokK17SS19WKFZR8h74inlMajKIlYj/EMwymwmgqd3rfVqgRzXFmR9nTKKKMWHmNr1SrSj1rwz6GGc86+OLQqBiJulUZ1jM5SMG3mEf8gRod1h5iGzKBJ/5bNKTMKhBUwXUocgp5zlpclCE3rNMPkrhCh+60eLQ0542oWhFYWIxH5lCZzjkaEpZ0w1WEBQnIRXpL16nIqsFABcqxWlqyjkJK+ImpjWxmY+emFOiTsajXYEpQpnnI0EAIAJcLGpU8cJSlyL1pzPpJobOiIFnSNWrddlpT7mSVHWObX9lAQcZ6pjJr7b1LcGoZ1/I2ky5YUh0AEDBOdy6V6p0oxVVBctcYalAApUSAKfka2KhstNWfUWwcIwc/4FoqYNbKtaySwkGIj76l8de8XMYckZ8iNmyy5ZWJH4VREE5e9WZtieaSJimaWULEsZyprMy9B2GjoEC+Nljtr/1CDE0a5vbUnCpBHInPIG7XIxgBky2jWkWrdPPhjLXuhPhT2M3U1z8ldE6GQTAIap7XfI6pDnR4a72cDEYAr1CZxotb3wX4tzzpJd3hLWOSQHQCvn2NyGy6Q9682lBAmnCqTf1r3zP6x/7Pk6W1hlqgstLoufWd5vqYY91wCGQp0q4vNmlUIPD9sP2nJEDafTwdREVYl82kTdpXWuKrUvhHImYaNItzF3zKmPm2ki7/7ExybxbGGws4bCo5DFwV/9cY1DKkTeTzVWSZ9ulL4UpyPOKbGhCO8x4kFbKlmWVmK48LVFahxnvi8I8vizbJTNpzK56cGF229s1l5bKARPzEoHJm/6lQbl1VmyYOfXmO8FKVryhLqAtG69pETpMxPpuixAxXkW3FWDjcjST2Mkb96qw0om1hsR+fKdM66iuodEvfz+9V5FRrNQncnJhDByBXKzarV0qBJ4xjcAso6VuxLB1WyM2MVfjj6SF2bBToRpsqbaaaK/+T2vRWrgTM9urMtN1sZUXVGSr1Y7WjqrbhhY2aPtQFs1Di47R4WVwc5R1cCt3dLKKliIfud1E9Zrj/hbv55i1MNFQjA6ifO//lIq7dfy2zanLsuXREjyl7+4cwjmDX6KcGQBpdnhKGafvgz+u10CZcxjykfGN3k95EgfMZ9ESDA8AwM8kfyjETw43DBcm0TAPqO44rj2U9yW3aAHvpHEOUJPjr+dfOW5ZOg3foY+zgD08+jp5gpZUN12e5xNjCaPOlSH/ZNa1tro4i651mq13f78OuzgLKFYZbv0mFEfTHY7CjLRXk4eQdbVu0JoGAFS77sYMoRJB6XaaxNkn3/D2N/5uzBfGEJaEl0nNiaJudi8ej2DM+uDHRWKi1Buxlk8lEgXfTMi/xMU/ATgAejBw0OdxlwOeFo570oz4wLbyrZci5ita+l90/z0t77NCOHGPR9Hz/kSlj7VPQj7y4a/x9SKF/MfFwXKX/7n5XUzmVQlfZqDwYi13oPT1i3jN9Wr/ToYXB3gZcXvxs5AZkpCo+R9Ni08BZentx/4hlHlVbb7qmUDRr1fAvyi6JtXiv6ODNKD4ugE0oohiuwPUNGsBCrRjQB06Kv4zqCVROCmROwxoBvarQPoZKAPEQJdAueQDB77jgK4KQRZywBK8pxq7qMP7gjpSvBb8oAuEwQw0kWPrCcrDQQiiKhLcQRP0HFo4GXG4BiP7vCAcoLAqwonqHVq4GZ9IvdVzwgfSwSjkQf9IOnGgPWkCwSxUHb8CLC7sQvSYt98DgP/gI8P6qS00lEL/8LeeWL43pB+42iw5jEH0UDjqk4N/wsPqMUMi5MP+iw78yoXvC79BVJ04PMQ5hI4sU78xdETFyaw9jMQ+fA6VE4f7u0TjQS1D3EREtA3JE4cADEXjeYZLCLBSlMTa+DlxmDVeWEXcES5NhEVO5Iykq5tjuEXVoa9d9Cnb4B5w6MAPDEbUAbBRI8YxSTiyScG+w4ZlRJ0Fe8Zi3AyzE4duqEEPuEFrVByVKJFsfKmJ+4xuHB0UQAdxvJxmNMecEDECuwYkAAAaQDJ3RBxsjEdt9IsTukLW00e2obF+lMeUo6EwhK2B5BsQM8iDrBABEQf3aUPhY0j/tmmzhwSpiLyQOSMD5rtItSlIjYRIsEiRX2i5ORDEkOQaHyNJqwILHkkZABCEhCgAltyajHzJjfyKeqEFnWEEnOQaKquwnfRHeSwTUBTKphE0oxwrrxClUtAZAVzKpjkGYHFKx+qKRanFqmSaS8tKr5irAIEGCvTKnmnKsHxKnDC0DkSGIzjLnmE0tdTKmwgQZFBBV4hLngFLuhRLnBibLqgjQACBvXyZYXNGv6wJmLqFXpAdFNCDADLMh3E2xaxLmsAFXLDHFxgCyZxMf+lLy/zLiYIFxWABCPDMz+QXxBRNuaIJXmAFGgCAEoAA1fSYymzNy4SJXiiF9xkBArAd/9v0F5nZudyESZjwhU5AmAMggCQQTn8xOON0TeS8BAMAgAtIzefsFpmTztF8iUhQmQ7SznfJt+5crZc4BJ1BgPFczaAZN/Oczl/4gwDYGfZ0F+6Ez9HsA/pMG/v0lo3LT8DgHDcIALRZBv/slrELUMe6grO5AQP4BwR1FmLAngVNuRgIAAHAAgnlFli4A1RYpBAV0REl0RI10RNdpEGQAAMQAECwPg7VFbchRQu1CV8gBRawTgtQBkuEUUNZu/ek0ZzohVawAvqUACi4kh7dFaz7BV8A0iCdCQOYDg9tEQFAgF1Q0iMSIRJyUiilCV/IhUbqhPdJmCF4Bx7NUjlpvP+e6lIv3c1bGBtcKIKrsQAGqIQ0XdIwiok29VJfwAVa6IVmYIY0oM8J2I4kxdNTKT6Z4FMLBdNGQoZr2AZGaBEOSIDsTNRJWdOZaNT8nA5aCAZoAAdwYAXeigAbqM1MVRXd+9InlU4/pQVeYAYz6QXFAAAmMB1VTZVFBSpXFc1HvYVI9QlmCIOr0QE1IIDi0dVTeb5elc5PDdUesQY+0BkSoATnXFZUYVW79NWw9IVbuAVZvaBv4ISWi4BImAUQuMlsrRReZctuNcpcAFdhBYpbiA//QSXxZFdNjYT4e1e6/NRfeAZ7OYYouBokqMZ9PZXs84pOJclvvYVcmNWygIb/OdAZFfAFNFXYLCE/sHBYg5TXYE3Cn+iGSVgLDCAFkNxYfvXXrvjYbJwOwRjYwniF9wmAO4CHlbUUhvU5eC1FP43YibW5OYUfvdJZSilAAPHZQwzTWziGka24NbgaHAC2o6WUF1TaZ/xUX5jZ0MAGQWgRD2CFlbTaQtnCv3hZLgRaib0gtACHUkCYCBCELitbQxlCWVzaEmzaY7CGs/oJXIAeAOiCdqxbQ4FCvOXDre1a60AGwRwmayjcSTnbzUjbn4JVtsWQaKBWvMKFRozcLbnb56jciuKFRuJbvwUKcm05DNAEi/xcOTlc0c3bZtpaZxAcDLkF2QSANqis1x2U/8m1jdGFJVjFhWQYJeQyWPDhBt8llEL0wtm9otJ12r6NkmZYA/rEWI1lXhmBRB+CXgqSUjj1Bds9E26g1L4LhRfd3i3Rw975XvW5XOMdnFI9G5xdXzlxXvf1JV4AV2KIBtQNDV+w1S4w2vuFk+79D+HlnfAF1GZQvEZhBrUCAKo1YDnJxaJ53xH7U1xAhosBla/VGQ+ABc+tYCNBraJcngymGf4Vi/+dlbeNW0ZQsxJ+k1Z8xRNRYJph4EC9XVCx16shg3ykYS254EdTYWEB1g42Fjt03CK4hiF+k2E0Yt75VGIQ1SUWB8292M6F4i2Bx0I7YibZYQfGYnH4hks42f/W7eIt4cdhCeMcSWKonZWavRreXWMtIUcUfpU3dh04tWIA9pFgINolYKs7RpIvdpUcLpo/5eEyHtbrvceqNWQsaWNOUeTeaVp6dWRxmNRKZQX1neQZGclpuWT/gNYr3mQpoV+5pdtQPhKXFJhShg6gFddU9olavZrBdWVKPgR5iWU+BpBMluMyJtar6QG62+UjIUqikeXNOGVAbpRprdYRTuZXDpfETGRgblhwrWVbPrxydapImOFqLhKd/OUQm9dh3uQf9p920F5y1o87+5tm3ko4FVhonpWCPdiEhWcaScuooWe7hFPM9WagqNiLzdh+LpKr9GW4CeiJSueCnjz/ND6KlFVoGgnNedZm6QDXe5ZooqDjm83Zi54RcCG2znnoJsWFoG3bjxYHXyBaLShgko6Rudy2N95bdS5oZpDaCZZkmoaRjG4dWVZcfC7jrw3bTwZqGWFN/FHktRValxYNuHWquX3npa4P3ITf2d1b6pVqogDcXCZcrBYRXMs27ancov7qsmhcY0ZmshaRpi6hj73cqF5r1NtcFOBiuBYRrTauJ5Xe071rxABn1nVdvv4Q4hyjNq3dHh5sn8jdOu5dxP6Q6LwiMN1g+X1soiCG5EWC5aXsEMHPEirdV/Bfoy7oZ5gD7E3o0AaR8lwiBvaFZsiwzQYK863U9HVtEFFQ/6f+01yQX3AAHdtGE/oV6d0GkdHWHhY+bTQZbuIW4FyeaeR2EJ0roTF+YMN47seO4Kn9aepukN4e6g1WYrfd7rUG4TqiZvB+kKdDIOZ24dAQ7sGGYaeSYfb2ECbVHuwmkPlea3YOYvx+EPEesUz2YK0674I+hiZ+YgF3EPemYj9GZTNK8FTW4tnZawdnkLt7HP6WnAov4zNO48PWcPwIPOPLiTheYv+2ZTre3cku8fzY1L+p4sU1FhZ3ZEG+GkKO8QbhcIBmZDLeZBxfYp6mTxr47h6/D3elGRX3ZiIHlU7uO6VWcv2YcZp55o+G8jMhVVO16irXj22lGIjtZpfecv8fwWXBHWswX3I3IhknX+szJ5BiVr23ZvP7aNZxyfLHlvMn21wSWO87vw8w2r/N42a75nMQDw3VDedxFvT6YHJXCVlNJm4pUXS0AHAhfvTwyHNFsWcbr3RL1y3PbvBNtw+e7ZSBRvRQD4pL94mDxnBTX/JWKr9ECdmnZXVkc/VuoGiUpQZZt4/3a9kliVmPznVdD42QvoNYAPZTj6thWWmCPnb5BnGYvhomAIRmh/RWmlEgM12dnnbtjtqreQEsWFdt/w6s1RG1Dvf+3m5sOF8HWAFMRXfkAN4E/u1Vb/cXG+76DoAMgIBzr3fuCN3jM12v3ncVmW9c0F0KSIqBT/f/SeCpE2H3hJ8jVYiBq5mABYgDiP+Oe5/lfG9pi2+PZDiFJigBI52BWvB47yh4/whshCd53uiGYAiFOSiCFrma4HCCcWh574hdExJf8p350LAGXJgEMsABndn5AJiAC5gBWxjpn+cOkO/F39bsoq84WGCEMNDdnccAC7gABWCAJKgFfKB6l/+rbgcM6W1ure+JbyAGVhCEKGi5ncerLngDKHgAZU2Iq057y0BgzpBtoi/6a+AFTpgDIND5q4kAGiCDUEgGdjEBA2iAwgz88Ghf0cX64913ZsCFS/B6vHeqHrgDWMCGe1CID5gfLM3878hf24Bv1P5qcEiGV6h7hMF7/w+IAkbwBXQg4de3j8H3C+w2gHC/hl8ohcVv/KtBgchPhqkXfg8pYsr902A9cFaPBl4Q/a93fNNHfdWffhE5Yc6YfVZnBljIfdLnfd8XB7IdfxGxYWy+CQ/f7G4ghuVnfNJHgTAAiFDJ2BkoaPAgwoQKFzJs6PAhxIgSJ1KsaPEixowaN3Ls6PEjyJAiR5KkSAwRql8qV7Js6fKXr1y0biHbJu4mzpw6d/LsyTMar0thaAQAYBRAhB53YDmLV/Ip1KhSp1KtavUq1qxat0b81koQrZdiXfa6RYsYNHA+17JdywyWoCgqjhrlEIURLmz6uPLt6/cv4MCCBxMuXBBapv9JvcaONeALF61ezb61rdz2G7FSc5BwoAsARZhQxND5M2z6NOrUqlezbp2R2KFSjMXGnInsmuXcO60FJaMjAt2kd1glc+r6OPLkypczb37xG6xCYWe3LHv2mVrdud8K6kLDs128ep2TL2/+PPr0WKFpUkx95ePIk7VfzrzZg2fQmnyRVu//P4ABCiggbKG8B5NMNOFGX0+8XbIGEMAdldQapBRX2oAZarghhx0GBp1071kXTFoM6vQWI0OBF4UgsGCTj4cxyjgjjTVuxJ4kizHmyy238MJMNyaKg1kpfESBH10odBEJf/3Y+CSUUUo5Y4E7Jngbg9fwwskcEQZHAxn/AsEzJZllmnkmed/Qcsh0L42InW7M4CLUd3RxUMRSL6K5J599+jkYe5Ho2BKPPgJZGWasxIXkUR5EwSQ6Tv45KaWVWlpSlS7l0iOWa13zi2ZAYJBfmMmMeSmqqaq6KkRqHgJLdT3+AudPQakYnFIu3sMqr736mmqggz52Sy6H5gROMq/ERYJnjjLS5K/RSjstmsgwYqBKm95yzILidGOfqPmFNhC15Zp7Lo1qtvhLWbfMCg5QdBY1Ya7Y2IMuvvnqG6A1mkTCCy7E/vJVFMzaeZcv4uy1L8MNO8yctZlwwgcZ4SYZ2mgYPrwxxx2b5osUIpAwr1ERwCDEGWrAYYfHLi27/DJgRglAAQIKMECAQQ00ADPPPfsMlTJOYKHHz0UbffRI/yiNNNNNOy1RQAAAIfkEBWQABgAsQQAZAKIC/AEACP8ADQgcSLCgwYMIEypcyLChw4cQI0qcSLGixYsYM2rcyLGjx48gQ4ocSZIgv5IoU6pcybKly5cwY8qcSbNmwxIibOrcybOnz59AgwodKrMf0aNIkypdyrSp06PN1hgA8Ojk06tYs2rdyrXrUF5lJBiwkc+r2bNo06pdu5YUGSABJqSjx7au3bt48+od2azPGjYBAriyurew4cOIE6flleaOAQcAnPAjrLiy5cuYM8N0O6nTDgMU1v3TTLq06dOoGTb7syZUqDYBBMRKTbu27dt7ezU2FYqRAAA3/o3GTby48eNJS5GZZKpUJhYALshDTr269essV6cJZcpUpjIGJsz/wk6+vPnzEX+tudPd1KVLEQBgQU+/vn3yyiO114QqBIAE090n4IAEmqYdd93xZwUAFhTo4IMQFhbMeu2ZwkkpkRgggB6URejhhyBelV+FnYTyCgcADGCAPyG26OKLQFkjyHYVhsJJLkUAcMA0LMLo449AsjQhezVqksscAEgAR49BNunkkxmVEoZ+FTp3SykYBJACPlB26eWXCslIY4Xe0YLLCwAgAOaabHo5JJndZfJKL2EAMMEiTLap554hohIGI3CaogkrvkgS2xD38Knoog5ac0gZnQTKCSq+tEICAApokyejnHaK3YRzBGphKL7k8oSOoAzn6aqsGsfKn6KG/9JJL70IAoADQ7Sq6660OZpGpIHayIsvpmAAwAK8JqssZsSsEaqopBj5iyxAAJDBsthmW9grZAAqqne3/JLLGnYmoe256KblaBmcfFvmL71wEkEADKRr771ZNfvst3L+4ssrUyWA78AEJ8XtIe4KSugvtHwBAAVbbFrwxBTPtG677rbryy+4SHIrBCtWLPLILh3Txr7fchJKL/C24kGKwkhM8sw0d3RwwrxxwrIvsCwRnRk1By20Rtswwi7Owv7CcCB2fhDy0FBH7RAybsxRCs6laIKL0riwMm+9Mku9FjFZ9yI2dTfjDK7SvdyCw39Pn21XMJKI040BsMhN3DaRHP+tdiayKO0LLeRacILeeKGCAzgGRDEH4rZRbbXa3rWiNMOmACDAA5DfJQ7j4KBwS+ep0UIGwpTzt/Evt+ByqYphk+4VOK+kgYo4spfGdxoYq62x0rzk0oWOud9lTS5IHFN8ZlSvcTXlJbLsLy2XJAny8myJI84cRWBvmemoU26j9AzzkiWyWUVzLzOmTDQ6Q8QUdLr2jHDAuPeH8U1G72qHIq3S4aqWmrLCDA5gozKfE4r2tMcMZ1TEGgaAIEHEcQ2GBKIg4mjGN3AnEHCQARkECUMrtEeGLuDvMM17HuWcszXg5aIPTRMNVsJggGRUJhQYYAZQFiiObwChFRUpQ/z/BKI9YnTBMQqBBicGAg5NFIITfGjDMwbyDC3czwDMmAQpJPGFKZ5wL7coQ/gotzbB0eIWseFcVs5kAF5UhhVy+EZLmlFBhixQIJEAwAgnkgsaDgQVaZgEBxiiCQcKhBHBGMgvlsBBAzAiEwXZxt2+qJdtSIIMmiBje/p1OVocAwWYiptTvsEuA7DCS3c8xh/02MiHiAMIv5iIOAoxEHH0oCBIsKFAoqGCSVLSMFRzgwrJqAnLXe4WxKjTAQQSO6RMIhd5uwQqtQcOPiSDlRIhBgquCBFiAIwgtKglEJpBkCcA8ZeFCeMYiUmpy+WCF6X4mFac4ZhE9mEg3dAEI6zR/41SGOB9AmnGezjhSwNAYxIG6MU2oOERcZiiEMkQB8BCwThxkKIQzABHKyRxu4FcgxOXmIT6DJBIVAyEE+E0gPJC0UoDfAMVk5AEMe4HC0a4USDW2ONAkIHQUMhxIwvkBDGYgc2ITOILFGFEQZEQgULc7RVSIYggoopOvFiSDJnQZHsutDp43eIYKFoAnrIyB3KSEzwCkQQz/tCFPjhjEhggYiSikFFi8GEguOiB8iIBBITw8K9/TcgrboGKHETCFwaIRBoM0IpclAIIgkBGKzgAQgOgAggzdQYZDEAMSHJWIB4QCDMYYQAV5IIgxPghBYsAsFaMjgM/RUUEfGm6u/8F4w8eSQYkiapTiExpIt8gLUGOoQID4OAQheAmY4Gg3KqqhRl3aMMwyRi9y/lCFs1AgrV4oJVeaGIgAYgCEQVyBw448A5o/QMKRnqNzXIWA2YzgANDcRBmROK++M3vfTuhEHCEIgLKM0AsdWgAecUvE1HghgFIgYHKgkMLBuADOOKXSAOYAgkG+MM3nhEAgCaDA+cssAqqaQAs/XQNfR3ICwiCVI5UU468nQgQ+CuRV1SYIMm4VAQOodxcDNK5dbkFGQKh1faM73IMOwYi7EQArXhwg9obrzVGBwQkDiR+pBUHMtIQP3EUIcUJxWJzL3LKNWBYIHkzpQHCIF6CWIP/A+DJ4B1koWZGFHcgjzulJjgwSXEgoXsD+W4bDdAFEwqEBvckSBqCAQ5m1BGDfw0XYBESiliKI8YHcQMZxmyAF4QYIoIwCC++4IxXoMAAS/gpQTgN5K1cNatF5s3/lJaLXvjia1vhRBrWwOs1kIAEbg7ANweC20hogjkjjV+GDnnnj7zgEAORxGfFgYJlD6R9BZ5EKAhsAHHoINEGqBQRgSAHRRpA0AKRAwAqeI0ImFS+/yTIHQBgAA7Q2CC00K+z9Htag2Sx25fWI0I4EIEDIiTNEFFiQZiBgzpeg1zoNkBlW40W6Dov1s3JRLgu17ZnoElg87SyQMSr6lcEYKQD//nCjwvC3/iierEHAazMW7pwA/S7cSSQIwhdThBDFmTnA4lEHeOHWIF8l+c5SLFsDe41g19ZE3DRpUb+UIZe1+kJa4jlz4e4EU74/JAXJEgZDE3xtQiZyBh/l3Vl4Yw0PMxpWrkDORWNxYHcIQcGoSo+DTCJdQukGxgghWX9LQn9Gv7eBynWJNstXE5goKCJjStBvsE4IP60G6EWyB9oUMPTQlKCJFWzAfqgAzz3FWC9sIIX78bzjNyRF0WFyBMi7hBoF2QO1h4I2Q0gi2aXXSvbuARW0+6dVyCZYch4xa0410ykwCLzBMFt0Q3Qg8cVxBQvaGQz5ACOcN5Pmsd4Bv99O5IGFHCwED2YJJsNYjYFCwQbbaigywLNbSAgIrEQ/AUADAmOJ8DcAFbWDSpQbsumAiP1DDigahrBQ7kQew9xByJHEB5gfQXhTQbRCkjATYLwbgLBCW32ewR0BxeXdoPSVRzjC8zwMirSIUyBC3eAAU9wY8xQCqW3BkB0DcJmEOAQBaQADuJXCAcEDkiACt/ACZYSDIzwdRrxAlZQCkWIBF5kANV2EHMQCN1wDazAB4Z0DSoAC+CACp/2BKYATwPBB27QDczwBX9wRb2gAtDgQGGwBsRAX7dAUBNGBje2gNpjDaSAVFqACkrIEK0AaDtVChf0ApzACgWlVDooB2v/YA09xAl3wE1hIG0giBVnR3wW0k4cdwvcEAXRcThZEQy4wAu3MHHNcAu30Da44FKcoIBM9AqXYArcZgDd4B42xFNctxE69E6SQAvcFAqgVxC4kAmkMHG7xAkyVRBvRWcE0QuXwAm1KBDEIAmhgA3cwB/3kwyaICeQ53racw2q6AuqiHIOcQ0k4HRYpIq9kAuq6EvfkHsF0Qt8MAd9cHMdhAJSd4lMsQ2cMHzEpzImODjP0HcT0GQhgiXfyBW9cAw0JxJ3QHsKYWMTAQsQxo9NYXEIknayQj5KQwvJQAwREAH10iJpkIHYAUGwmBC0JBHiEAX7iJFIgQtDpok5wwvH/4cLvrANbwNyIPIM9aMD8og2uMUQ0bBEEhEKCCWTSeGPAEl80ZILx9cLuNANSGIBBGAOIdINxJAMIVkecsIQnRCIDBEMd8WUSKGRNukeG2ddtDBlmqNGaFkS03gQUQgRdTmXP8EYaGeTmUALx8cwyfAMr6OXhoke/hgGzLGWnIRkt8Bow7NMhzmZ2MEMfNAaa6kwJggvVckJ1kOZoEkdfJmZFmIKmzk41oAM5xOarFkcnKCYpKkyHvmRzOBDadKauFkbzXCZG2mTR3Z8jwkOMISVuVmcpsEYfUCaOSOVxxc838ALaSRKxjmdifGPi5mZWdNCSDY412ANoKQABmAU1P85noexm5ipnBoXmAzDDOLgMAcAd+QZn3qBnMoZJ4ETmMgEDpnjACDTfPL5n1zBGfVZOeqZC7nwDQV0LNIJoAzaFX0xJsqpOoHZC7RwDeCQIxnAXQ26oV7BGERSn78TmLTQDOJwCEzGoSi6FcpxncpZXfhJDOLwCxEgAPWiKil6o0pxIAOaM7NJa73wDdjwcQZgozhapEOhGx9an/7DnFN5C9sgDm5HAQbwDkRqpFbaEyu6oxmnndtJC84gUcs3pFc6pj2ho1qanurpSZemgmTapjuhHkk6oI15fLUGDjtoLUfgpno6EyOipYJiTBPqieLgMRKAkHt6qCzxDH8AoTv/KqGBSZDiIJL0IqaIWqkl8SZ+aiGlsJmdhAx205OWGqok0ad+6qKBqZPgsD0PYwABIqquuhFi0ps7+psTWpXicAtx+aq6qhGYmqlLqp7+IgvQIA6EmSK7eqwWQap+aiXAKpjasyCSiazS+hCxmqmbBJjAiguMJg6eKQEf8A7TGq4L0avWOqdTaauqGQDoI67sahB+4i3WqpnAipo91AO32a74ilOH8Cvx2h2TwqmdlFHiwAcMYi752q6g0q/+SirNGpzikAvRWaUHu6uvAq/x2pHN6pzi4J2hNLHh6ivA0q9JM68Vqj3uuaAe66r6orDdEZXNup7aE0+4Ug8pi6zc/2Kx/XoJbameDpsMxrKuNfuqF8OycYKtwKqxF3qb/hm0ZGoyKKOwmmB8zcqd2mMrEwAHTPuqaUO0JfiytMCe4nBrApACLJi1ejq0RFuaAHtMMCoO2KACmLIOS2u2OOq0abuwPYpk7wRlYLCqdFupW0u0tKqebXMN2tMKt8Ihc/u3G1o0fpO2NsKk6skzJHppkHED4Mq4bio506Ww2fmyrOOQn4Oh4aC5bhq4aYumzVqn2jMIcTELi2u68sk3j3u3gAO6bfOkkfobasAPsSu748m5d7tJgEqyX2o3IgAAK5C5wFuk4DO8+8GJzUoLnqo9V/Aw3vC7zVuc+sM/d8tVoP+LC72QqhIVG5+wvTiaQtDbHaYKrG3TDdrjDJABBeh7o8+7vjwKupCqPTEQHTRbv40bCfuDv7ImucAKkgv0BhEgAdSgvQBMmeqLvywEuidIvrYwL6NQtg9MnepEwN2hsxRMlfBrNxugvA68wXp5VZnkweY6ubQQDQskBAySKCgcn8gwB8LkwX9KwYIJcI4QG8pQw/GZiTrsqC/rsOJgCwEAAIQgxOP5ajqsqWuLZCKsPc8gAQDQBBrsxKAJXdIVxe1LspD4OTAAAB1wDyfMxb9HxDo8uNMLtuJwBXFBDWqcm8H3lB78qxSMxI6gObEgnnUcmhbXuevLrBSssQ8bGGr/0A9pHMhVBQs1GcVFy8NUe2lYvAKOHJrPEAlfYI2u8cmgHMqiPMqkXMqj3AmRgAoGusqs3MqtcAzXcA3PAB0XkMldLIKy6sFdu8cz9TlSwCDGYMuTmYm5jL//ysMGCmWDcCuf0MjC7D1OCWvFDL0CScklKw60EBhYsMXPXHZq2R3TfLcYy8Mjqj3HYCwz0KrdjJE02Zfg3MacgJM8jEzaEw0oEAAIUA7rLJPRDCfhzLJZY8CrywsbBA5B8DC7ILH7DGTfTCb/rLAgzMO5qz1IcrX+4MwLjTijGSsE/Jc8PD3HG08CMAT7gNEZfTaJyaL+jL9R+9FJpj29MC8LQA8m/33SUmOZ5/ktD52pu0zBrNsMl5IA2WvTrbbRCbPTWhqiIeykdrMDxEPUrfaaKq3TwyubH80zz6A9dWIBaADVzmWeSP3OkKszLk299HMrQ6DOXn1C9KlJYU2akevSqIrNgcEA67DWvySgWvXWa/m5H13FYNWxeH1CfZHTbk20qkvBgzOs10AD9zrY+OOhHMmyt1vWEXWnFDAFkI0/eo1xfE18mVC8FLw1qTpvEwABar3ZnVPYn73S8WrEIWyrIv0AQ63asoOka9naRQa+V/2WMSrTKGvbZ5OluW2tYQy6IGkAQA03NS3c92Kmxe2nbgy6wdkN1RKtzo04uK2cui0+s//Gw7xgq25HnAqd3UKjrJnZ3ViTCVyq2CWbR/y5DM1t3tkC3dytpRFd1uyJRhEAtPQdNeSqpDvawtTNaIENnv8tNugt4PVZTC7tQt/Q2I+d4EOjqIw6qw0uvdZsoaAopRQONQGupeotKZv64DA7bxJwPR8eNAsu4qR53NQNo5mzOSseNNWqsCNeI/Fs4skcDMaiIjVOMyFurTkOzt8t0bdwDc0ASj4Z5CMjJVQiuDZpyC59Xc1g3U/t5CIjJiEr5Zro0Sb+VVDKIIaq5RMz5Dj+5VL74HXqMQ4gl2ZeMBWLvyPe03/tpLeAa3FeMCBLwOp9zA8+OM6ADCiC4Hs+MAn/m8cYV80mTr0SjgDweej2crNRrNvj/ODi+w0dXuaSfi5o28ZFNrIP3jbcQLAp3un2srKSzBtaFdAmPj3PMONwjurZQumrzuqalN8PTr0+bqy0ri2fvup8DeYmrpPMwOS/ri12e+tGRkxrPuq4cA05gt3Jniyoe+thbedXLQvRMN6cXu2tEuzMjusJo9S7ngxuPuvgzirLPu41gjNW/eraagAyPQbrvivX7u4PPd0SjQvH8DKGfu+r4rje6+5iHSs38urTwwyOfS0CzyrCa/CuHSh+/eoguenDUN4Pzyb5LvHkDieJ/eCPaeogo/Eb/yW0W/Aef/BkUtmvHjyooDlg/3PyixLxKz/xFRLaCj895uPrNK8o93vzwQInsL3rwYDsP88n3Sv0VF0hvP3qjzntSc8nEcz0Q98eMI7MueDtU68nQW/1V5+/Cj84jBCm893196E/Kwz2HF3AO88wXqOuaL8mVc/2Qz/BO6+KAD/3YNLBdu8u3KHrPH4LDS+KfP8kKvz3R03gf00Lm67Ph+8kXkzIil8hkyDagU4LSHLqke8kbFz5oqIJJb7z1CPznd8kUAz6KUMKAu3SjvXjpx8kk6/63xI9eSvRrOAfIHf2sU8dn0/7Dh3PvnD7+vsK1z2lJt/7BHLHsAb8/iwtw6/wDdM06ZD8yi8gg+z8FI+m0f9f7PP25tfvIr+v/ddqXcTvvoYi9+EPItuQCXhM/pv07N1/1aVQ6OsPItkP/2Ri5/PPwwDR6gWAC2MMHESYUOFChg0dPoQYUeJEihUtXsSYUeNGjh09fgQZUuRIkiVB3iITyNRKli1dvoQZU+ZMmjVhckLl69fOnb568QQalCcuJAAoJDGZVOlSpk2dPoUaVepUqlUNbONEJpNNrl29fp3JKZTOoD6Fnv3VKwwACRDk+bMaV+5cunXt3sWblymzO2tCgQUcWDDMUJ1+CjWLFqivQAAEMDCnV/JkypUtX8acOSGulIM9f/4aihMvtIkV79RkAMAAA/80v4YdW/Zs2rD/sWoFnVt3zFKacik2rbgVCQAJDMCtnVz5cubNnV/k63f3dOqXbp0OfjYXDgAHjj8HH178ePJ5eaVRSV197ku0Tv/KXjYKgAlJ9rkun1//fv79L3IKY5L1BvQsk1feg+8wtNwAwIEh3vEvQgknpPC5ZviQjkANwdKEFbKwU1CoSSIIYIHWKkQxRRVXvOu8PjaE0StOTPkQRLRYwQAABbRhsUcffwRyJAAFjLFImsQK0UahcAmBIAP4CTJKKaekEqELMzQyy5dES1LJxYAwCpQqxySzzAld1DJNl0T7DUGe4vsiAAng6MdMO+/EszlSyCBSTTV7w8XNxZIcxAAB1Kgn/09FF2W0smb6wNLPNDO5TtA3QywlAgBukKdRTz8Flarz7pBU0kxksXTQoRwAYIVQX4U11pH27LNULTNpJVVV0xLBgAPSkTVYYYeN6NFIbTVSk5x0vXSnGAyYgBrkiKW2Wll7SYNUZNPkpJQadTVLDAMciAU/a89Fl9FS+Nw2zU5C6TJVn/AAIABIzE03X32rbOaPNP5qN0sumS1LFAEAOAPffRdmmEVstQ3YyFB8IzgoWiYwwAaFG+a4Y//2jCTiLEvJJNCKeerFAgBGuMdjl1/Or99/Rc7SupNRPkgCdWDmuefmglkDYppjNPDmnp514Dufl2b6tXVDHrpITXI12v8XpJVpuWmtt9brGX8BjhpGZb8lmIhxdZmWa7XXrgpoocPWsFuymTVbAFc2ZjtvvZN6Gu4Y341XVykMBSTtvQ9H/CNrBJnZ7w1FI81oMAz1A+/EL8d8Ircdh3HiNm9ew4AAsLA8c9NPTwiVMKDmnECSK735DtFvMBx120+35pA0Oml9w0zcu9kX2QPg9HbjUQd6jt59PzD4PwIIYIC3jqf+clbCYGR5DTuc21JfBFFNgXWqJ3/v3HfXnkCcuhfUF0TC57F8+dVOPn31xwqeEU2Nm79/rV8hQ/bst57CBE5QvZhEjvjnPwbyLHdlmNEA1QO5m/VCEzlCQAM1+DJirEH/eRJUT28+R7BecIIDAMjgBlXYMAAKEITVgR3BeBEKDwAgA3RYYQ7z9cAIvnA67blZLkpBnAt4QYdHtNYx2vBBH06naCfLBSpQQBCkINGKwgLgIZpIHe6dDBetmOIBfnBFMsKKh1uczozYhyBcwGIgB3DB9Mo4R0YpcQ6lQONukHSyW9CCBkY5gQHqREdC4imLedzNwCp2C1zowCgfSEfpCjlJIG2DERBEpG7YxEde9AAAFiCAOWpHSVL+CBluuGMmc0MykzHLF7ToRRE+CYETldKWQDqkKnNDqYq9MhhFsQAtbznMHm0jEpjUJWgyAYteyoIY85mAMIk5TRSdcg14/0zmZ3DVTGRYgT60HCU1xZkfWpBBi9n8zNgI9spkfIEtD9iEJMc5T+c4IxJk6CE6BSM3EtKCGWR4Jz0Fmp9gpCENYglFQhW6UIY21KEPhWhEJTpRilY0oZnQBC5ysVGOdtSjHcXFK5KRhgY9YKAnFc8tyvAHfX5GkbrqBS2gwSAHmBSlN2XONiRBBk2AraWA8ZwMb2ENPjSIAThFanKsiUef/tQrrCRYLm5xjT8AAANHTWpWY6PSc5qiqU7lis2YlQtcYEMSjsGqVtV6GZ3ihiVfBStNnqgrjXJjEo4x0Vr1Shm+tAGbb41rV6ZGMFz0ohuhAEAE8rpXxuIFJelZU/9gbaJOXd3CsKSoF2sau1m6tHUrhJHskbzFrFv44husgJ5mObvaqvT1r1sKrUwAxyxaBAMcrdCUAgzgDdb2NiqPpQlcY0tBedGCGOCgRW59u1ynbOMSbp2JcCUb1OIeAxy32B9ztauU6LxWJtKN6+vAJQtkgIMXCtxuekkCXK6AF6xAlJcskiGOXiiwAurFr0e2kQno2sS9P51r+2TBDHEE44QIaEB+FayR7oYmtizpoqVi2gxxEOPACV5whivCGch65b/oXF+qYuoMcSCDOBkggIZVHJHbfBYsH06mWNYolJhGo8RELMCKddyQ6MAYtKEtYKp4MVRxMGOKF0jxjpX/jJBcoMczPkYkcQUl1WsUWQXdSfKSd4yVMFzCpZIVYarIig1xOOOPB8iyllfcY9BAGY1iFVRdxfGMP1IgzWrW8Hk6/OTAwldQlu2GOKzhSTuDQp54Zu6QEhnXACPIst8QxzXABEqlIVrBV3JzTTINwgi76TqQ3gYwIbAIS2cYTdTZ9ADVaKnagkMc3ZhPMEudYUVP8Kd7dNMrj/tqaIJz1vjF9IBSrb2XvueV1hUHONwpAZOG89esPbWw9bnJXJPX1coO6LPVS6vHoROq1Z5vsknKbG2n11jD9jA6eZlrf4oj2W0oabm3O6oiodtxyzwgLSjs7jsYVd7a5Xa9k7lN/zeN2N3i6ANa/73cfh2r26qk7HtqfHBBoBWHC2ftw9Jk76jxE0FDrrK7zyqAxWKcs+uqlcQyOVsEUfngl6hXyU3OWJlxXDA2j1ixFTPmgyO2RDPnrMYlhfN2Ufc9cnZ3KTIL9M32rVRER9a33wPog7citUyn+deQBXVbwfk0jz74K5SLdb1ubltcN1Xzpv4LSLsbuzoiu16dfvYmDhZBrT742xcY96Q+g3Fo5/MLI14a47ra3fUtDt+1avacv9DjwKEFsg+PXsUnde6NlyDLgWPtgxsYhZWu/EAX1ziaAb7eozG2fA8ujmMcOPQ4ZfzQTN85ip0mpgQ+ODJqeIFdHP/69bZUHeuiNnsNidf2+l49M5rEe9//npLn453fiE8gP6PF4AdXfneG0XznF7J+nJv+ehpN41vY+ODO2EB3jMH97tPxei50XPi56KHTgHz16DcK6Ns/TOgvT/56XJadm6rViwYRMApj2D9x6iAm6p3/yw0ZOw2eO7gCNApqSEBqaiH7cUCXMozTQDp3swYLCAALsMAL5L9DQKb02cDBkLKzoLoJpIBPGodBMsFSWkAQWsHAkDoX7IW2c7dkUJkNUAcoqcFSykAcdByvE4pPWz1bYBULGMIiJKUzeqEcBIvqEwq8O7hTOJgOiAcilMJCsqMtskKvGL83KbzVoxcAsIH/RAnDScqlJipDrui0sog8w0s2MagXKKDBN5wjKiTDjqOR0iCv1buGIHAMNehDPyQjJZID7/KhOTwSeCHEcHM3ZmAB+viEfWBEOorDPJLE7+KEwOmFAVu9XjghCogFMOxEK7KkFIwykaE2GkO+g2MFTeG9RWxFHTqlVNKlUHyJHQSK60s2RgCA4lgGXdzFFSqnrvrFiFk38jO/VwOo6AkHZ1tGBjKmg5q2gMG3s7A/d4MGWXKAFMhGK1qqlgJGliA4oXA5d0OGgZiAIYiMc9ShZnSqdTSFweOJjdqGvDuhAxiC8bHHFTImfAKrdXw8oPjATtCUDDiCgsyhdExIW9E8/6B4QXAoqgBQACOSSBXiqul6ulHkwba7Bm8iuY9UobbSBCAbutoLCiYsMkciN5XUIGSYg2saLj8Rxp3Qwl4gDgqAgFqwyQYKyQebQyWEjzQUBxzpjg8oSm3cKRfbyUlROzSUvLtCoUCKyv5xrQcDLC2xu8WghXDrBoCKANXqSvlhL7BciRzkR19QPXGAhqJwgLRaS/LxLLdsiRVcyLQwxRKTR5vKS/L5Sr4My7+hxGGsxVsISF8rTOppS8R8y9OLHJShBRITB4e0oRRjxchEHefqL8r0qhgxup3ohfJLto3ULdCkngYjzb6EEePjCfs7SbyCStc0nsmMzcrcECwkq/8qYwZPkgC81M3TabHehC0NaTR/LLApEsrjvB02U87IIpA6lLNWyBEkMwjpPB0Oq84fW48Q4wmq08oU8k7TSc7wXE71gMDyNK2zTKwBuK/0zBzqZE/rRLUO5An3+IZoKAoOME77RJwm27P8TMxEQr3yJIZviEcA2ADCJFDE4TIvQ1Dx1A2p0zXzqqEOkKYJPRxmwBB91EAYQsPyCoWHvDMQzRs9u9Do+iHgWcpkAIeKC4C9Y1G9qbUXjQmomyu5ZIbbxACZy9G1CTYe/S7diLBSZIbhZIsBLVKuiTYk7dEHHMS00LdgmCJZi9K8yYqUo9L2HAxci6ln0E6CWNEubZr/Iw3TJH0ykvyFIYuG80xTNV2aKW3TKmXB2pPTNZhPDLPTrQm4PIVRwfg2smIGuyTSQPWZcyNUTRuMddMoYhhMRt0aentUSA2Mb8QFXsCFGrIzS9WaQc3UQgWLbTIAy+IEFRVVpmk4Es2mTVMny2qMG21VphG6UvUvsOAnWELJBfiAarjVnkE5Xe0KNwOcV7oF4oTSYeWYmjPWY/WwUXwlMJolZ+WZXwiaaE239vKNVwqFHEEzbIWZy+NWTa0JVnqlxkCh+iTXjoHWc5VWm7AOX4CFtYied3WZ2JPXXa0JA+kFViDHRdVXfTHXfkVXmZiaXgiFK5sACS3YfRk9WK1K/5nACV6QhFRMMWyMWGLhV4Rtr7AohVy4g4c8gXDo2IUphdUBWaCSrVDgjGM0DjlKWXSZ2JZ1WcLgBFgoCpIDBX7g2JqNlY/F2ZDdkpYciOJchs8UWmpRHfgr2m5tCZK5hBqygCFo2nR5oOiL2hd7CQPoA3GFg6xFl+/r2pxlxzAIABRKBbI9l/c725triUsoAtGZAWF1W2rpv7gNDLCRhF4px07JW2K5Qb6VW1M4hKSZAD1gv8HNkyM03L41BTbIEQqYhaB13EUBxMgFKtGxoXTA3MzNE1/wIM4djFJQDQBwAqAVXVkxhS4QBIuS3dml3dq13dvFXYt6A4TwA3to3f9YeSALNV1eDYKDmIBiaNzfHZMxpNjwnJg+kADRaQJgUd5XyaXmjc1S4ISekgJNmYBPqN5XecUewl7E7ARN4IRX4IQQgJ4WIMjw/ZRefK3yRUpNUBYDwIUyWFsLAAQ3hF9P+UTfHF6W0N5MCAVYsqArE4AEWNr/9RRjgsUENd3zTd/fkEsyWFsKGAJ7CF0HnhL51VPOnRhlwQWywIVJUOAESAVl9GA7wUc3NVxOMGBY6sdXSIMMHoJrbOFF2cZ8wtCopeBXGKGY4oQUxqEO3mEgoUhTxdkRRoUSXoxbkAU5OEYKeIBhSOJFeeGERdjtPeAk8QVcoAVWUODWzOI8OUj/H2bic6XgVhihfqQFXyCp7oDYMzaTJfZXbh1hUyits4gpX7BWBzBjO76To5RaXfViWQgcX7iFXEAGP+2ODyXkMmFJtH3UNn7jKL4FZoAF4hDkScaTU3IDSDTaPN3jPlYMMT4GZ4DkA5BkUB4TQ7ZkJE1kA9oJXqitbWCFGpIAtYTlMdlLFqTSEXZjBIkpXogGZ9CCY7yAV/7lKTlMYb7QidkKVMaOW9jkpuRlX35mKuHNw2XPWo4zWkCGb2iGZSYIZ+7mIBFNqgw85STmXJgxOCYGbmjKE5KABSCAOFjnKoHN3KBfSSEFGebjecbMXrCGIluCY8wAde7nH/nmL6NM/xkOBViwZcRgJGcAB3DghBzZgAUA1IeOkna2NbeMZ4N+E1y4BRoVh2Qois7MTZGOkn+ejoCGkYHOhFKwZkvJBVk4hkDjaI9eAHeV6SCJaE0KLRlO34tGC1wOhpBDhqIIABQrailZT2lLSPt1Y5QOipjKBWhwt6AGABFggJCuah/BT6zWpwLWaa4uC2xmBsMjBjCJgBao07NWEfDsHH3a3k54Bab2QHJuu2/QHwAogWbF6xS56odTpRFmBSg+Gae2Z3cjBk+q67tO7ApJa8ZGo96YYbf2Y8uaRsLWlMPObCBxUYHJo+1N38s8GUbGhWbAw2DIAatKAMw+bQnhMjDd6/9ItN/HBu2zCGPBPrhuEITSfgAMS97cDg8RdTjTBKECPuDgRotcMK5AO7he+CMMSACzZm4USW01semZCGLXNpo/Djl34wY+yOA6/u4K2VHxJjb7fWLqJrxc0OjV44WB4G7vfu8JYVM/GW8CpmgaNhpNjuvV24aiMgr3/m8JwVMBd5wgzuQDt+6fXj1xyAX+VgACMIMHT5EvLbqwcWLIPnCgcOr0drdraGUHB/H+CHBbwV7pNvATZ8yvznBxoIUp+uQXB+80eBHMaxcKt3GM3mQ8BMEW93EUIdUR3zr6NvEiHwriznBY4PFBXvIIeRTSE3I/8eIal3J6nmwCbGWHzvL//MBU2fMTIg/zroalhM7xV/Bk1ti+M4+QYiVxLSlx+/aejEZyd3OGLjhGVzYAfLBz/4jXPC+SLwfsqhFjls5xVjghByDYQy+PXIUb+cPkNq/u685xcWgGb0pnYWBaSyePgx0+AjllPoepOFbx1ZN0tliASjB1GNe61jE9ilZkTi+NRtbMHGcGRLShLQDdWt8PbX2b+JuOTed14YbrPw/rRsAAOTHHfkBiY6cNVJc+gLbfgm52tFBl7M5xZIAB6MmAWdA/bA8Pr+HyBvwMcf52cMzlT0+2N2CVCVADwVX38iBaXL85ra7weMdSZKZ3cVAFEQyAFhiHfdcPbff3r8Bp/28XeCPHvU/3BRlY2w1wBYbPj5tFwq6A94lfEirP8VyIgbUVAC7QB47n922NRP8CeFY3GuuuZ3rnBWF3ACfYGZY/dTIQvo+PiYjfaZFnTIT+dHB4hR04RgmYgXFYbp7PDI+Xw5igaIsm+l7HhfzO8G4ohSujjxlYP6gfj36XoKY66asXbpWGdEO8hCb5JNbgRLEXj6fNJIDB6bZG++p2JnE/OL87IaNYAFCoh2uX+6jXHa7No05Q6r/O+6aurVcvsjnIkQA4gKOi2cJ3DrMFxUyYBOBufD+mBRzPcGIgA00JgAtwccxfDrjNI8/W6YBvfEY+8gwPBrV1DAQwc9VPjv/N9aG+ZnxfMO/P/wUxLucM54X5aJAEyH3dr43ClcPfNnHgF/7arK0xT7ZX8CS26PApYH7xgNzoLnCykX7hT81emMZXK4WBoI8OPwjC737L4H37Ye1XCP43qf+Jh23ZPrhruATiAAgAFgYQMGDwIMKEChcybOjwIcSIEidSrGjxIsaMGjdy7OjxI8iQIkdyPNZmjqmUKleybOnyJcyYMluG0qQJFS5fv3by7PnLFy+fQocSLTrUFy5ayL6Ja+pMEAcAAEYsKEjyKtasWrdy7er1K9iwYj2+InNoJtq0ateW4pQpFC2dRnkCnWv3rs9ctIh1ayqO2RwMAALIYDD2MOL/xIoXM27s+PFHa4fKcFpr+fLlTpo4vQqKd2fdz6KH9op7zS8xMhEARKjxADLs2LJn065tG7JJlJh383ZZ82bO0aA9Cx/ti1YuZ+CaBgsjFUMLCLenU69u/Tp222XP9u6+u+1bWr2K0yVO3q6vW7eYLRfHK4pUDyukZ69v/z7+/PohbmNE2TuAa2nGWS7n9RSagUbpdUxf4LzSg1QhzGDVfhVaeCGGGR6GjBtzlBIgiDH9hpNcCQ5nIlG80BLMad2U8oJULMygIY012ngjjhJtFyKPLbkF13goHmiekKXlAo0411xCQowDaJAjlFFKOWV920TyX489DvhKgUIKhSCK/+mtB85TUQEQwwIaVEAlm226+eaGHX6YJYgjBuflUUQamNRSgAlGGANqwjkooYUaqhEtZtEJIieaAImnUWCepxcx3KS2WgQ2PLDmoZ16+mmnVpJR2aLdDdhKl5BGqudopfViTTBdPBcdASCAeiuuuULJ4RpzloqZnSWquup56eHiTC7wASAffZsY8I+u0Uo7rX2JcvfrZY0+OixeknabVDIPRqgAAQ0YhA+16aq7bmyikoqtWqemym23rM6lVzChwAiACARxyi7AAQscFq++wivTb6bcIiy99YpmZCRMCjTAwBVbfDFJt5Rx7cEx/ShLkA0L5y1R6b3Ch5kUVIUxy/8tuzzRNpKQoUnHMckrsoEk+5SeKWX8eYBhLws9NNHIzOGGwTWrlPDCOCeoM0+4cKLFagFc8BrRWWt98S1kBKK0j46C7HSY9v7CiyRFSCUAAvRt/Tbc6cZMRiZgL20TqmR7SXIvkgAhlQMJuB034YWDyswdbSR9MCluKcyw3k8T6cshKkglgQIfgGKAP4Z7/vmgXX8N9schR743cb3cEYJUEwyQxS77dA467bVDuc0ldCv9G6qQn172L7lc4YABExvwTj+2K788jYj32nHjdTf9O7e+UEKEAMVTgMWzzHv//X6id/wjLKZTDyksCFn9Cbrgu//+dbjrji3vufh+foL/vWRCRAgHRSBENmYHvwEScDbOWxyPoleK6eEPRb3IhSnuAITsAc4GnnBHATOowceIr1SN6sQrzNfAkfUiKbSQRBhoIJXBWGAFn5CHADcowxmCZRucmF+WfsOKO43QOLm4BS3U04pDWEEwgLvAAkCRDs7FkIZOfCJJnBeKLJVCE5lY4P16SJQHqucWuPjFMVpxBx2sMAAUSAADDGGAfiQPim58Y0hw4bUsfTCEWkRPCdWDHGIwwxrH4EQUVtO6DBAkC8KAIyITyREb4jBAOuThHXf2wy724hjOuMY3sHGLOaBghQI4o2GmoMhRktIiUgQReOCSRfwZ4Bdc7KIvkgGN/220JxmXQIIgATCBDKyslL78JUR4kYbRAcgtnDHb+XyRRyDmgo/R4IZfkvSKNXTyOSNQwAPMBcxtchMhNgzDJAAUrDsCBRdd5IUlMSkOA/gFHL9gBC5XKIIW9LKb9uQmM/iwhin2JpVx6eErgxjLZ1yjPdEUxzNYkQaJAQADRJgBBCh0z4n+UpjE3M2WkCkyZU6SFrjgIzSgedB2vhMIAZBKAGDQhBRIlKIuLSUnwMmbcZ5OmeYEIjqbYQ2mjDSazmBFGMy0LCNAtAHafClSR9kMffLTMv4UYcN80dGFIeOSBu1pU77BCz5ACKUqTYEBjAoCLyS1rIkUZh8ws6V5Nf+Mi7TwaDCY8QyRYvWgyeBEF4QaghpMKKxH5ZxZAwtFUpAhnGuhafVK+FZa8IKP6qzrSLlxCz6QUSoRgMEMGFABo7pAsJ59YzP6sM+1/Eg81fvhW6lqVchi9a5dMCK/bOCvgqDhs7aFojDvEK/NcElVD0wtLuI6V9bWdRuT3VdDR7ACw1TgSQbQxm2jS0PCGhZhNiGRkGwaRMbycafEhawt4zlIghigANI9rxNDO1qZaMu0T0OtenpR1YJ+F7LXmGY1AfDJBGANvf6lYS/SoFubbSZv5/ltEIMrV25ctb49JQY8czkBBNTzvxbeYCkKC5MRMfAzNtVjM/vIUwdDNqH/YODASfV7AAXQZxgXfnEGm/GHNDSVJe2FalGkCsT4prPBJB4pOJjhCSGwoIwWSEAKamuA9sG4yfAL8IBZcrO5tPKVXvyFLGn544Ne4xi5QMUl/hCGIIjAASk+4g304OQ1ZzDDkmAJ01b5YWZ2ty8k7gYzfPGKThzCDVYAggpyucJBO2ACI+DCJ9LBDzYzGn4ypvFKtDW2o0xSPei85Ijryo1k8IIVlxBEGqKgA4YOuowSmAAFLoAABdxADXowxkH80cRG01p5wVhDlKfcEysvLBkE9TGeg/EKTghiDl0AtKBLDQAObGAEF8hAAgawAAZAoAI/SMiia63t92U4EqaI5zNoAvpRZjyzKdEgxi1KcQk+hAEJNCC1sjEQAhgQQQgrmMEAGPCAlm673zJ8xoxDIelelFOPvAhGaYYtiDWImgRnVvayWRCEKzjhDFBIwRAGZ9QGONffHp/hredgE1bAghWaOAQf5kCGP9MAthDHgAhgUAN7z+AGD4joQbS5Jn5/vOcyDIYRaMACFqhAAhgoHsT1a+gOICABM5g2zs1rgApQvQH/8jnWFQkAgxAa1aqO9gJuTqE1NeADWT97N/1RjCY4AQtqAAQcNmeQWiQkHOtYB0Vmjfa9y9Ae/PBHP/6ud74TnqIBAQAh+QQFZAAGACxBABoAogL7AQAI/wANCBxIsKDBgwgTKlzIsKHDhxAjSpxIsaLFixgzatzIsaPHjyBDihxJsuCGkihTqlzJsqXLlzBjypxJs6bCblIMCDDAz6bPn0CDCh1KtKjRoyqZdRFI6B/Sp1CjSp1KtarVp7SiODBQg57Tq2DDih1LtqzZot0kIdlhwME4e2fjyp1Lt65duszSREkjAIClr3cDCx5MuLDhkFm/9JFgwMY/wIcjS55MuXLctEjmBCpiYEI6f5ZDix5NurTMvFEECeILYJTp17Bjy57dEFeUL4cE8WG8AjTt38CDCyfcbdKSOYcO8YERYcK7fsOjS59OPSrqQMn5zAkQwFXP6uDDi/8frzKXFtzZGW11ws83+ffw48tnWByJm+TKI8EwQOGd+/kABihgddfh1wcjc+gUy4AMNuigbL2ch98hfxwyiQMAQPHfgxx26GFg9d2H3x+BcMICf/JA9uGKLLZoFTRtRIEdfoL8EQoYAQgAjIou9ujjjz9F2MWEufHRiXoAYAEdkEw26eRKIRKpHCeasABAB/U8qeWWXGoEo4xS8qHJjW0Zs2SXaKap5kFCSqncJKUw0lcd361p551bcoLEGm7yEYkpmYgAQAbt4WnooT1CI8cTfbh5oCmaSBGBBNQgaumlHEaohZsUClLKkX0BciampJb6np58uhmIjaRoImgL+Jj/Kuus1CnKKKc1hgJoThOMsyGtwAZrmqacHlKIkYBGwh0gwjbr7GilPJEqp3xwYkonoYQQQG/PduttYdDMcWuxfGRiSigGMGGABel86+67cxFb7JumANoId5/8Cu++/EpVyhJpzJtfvZqgIgEAM4za78IMExXuuMU2Wi8npQQBAAXp8NjwxhzHFIwWmwr8h6fncjKIgh2nrPJL0QYs8Kq6mnIJKxhCUefKOOf80TV3QFysIH3ErMkrzB1ggMY6J610RMSAXIjAxlY7sSl/tBUL0ksHJs7WuWQ9XstQZ6dJvaFw0gqGWHgd2dYGMOOBOGpTx7PP8/JxSb2A4gIDABfE/33Y1uJwEgDcfg/XtBZPh82HJHhnAgsbAEhgZuGFifMKMwAQTvlvqDzhsuKM4M0JKqj0pYbCm9vVDCzfBJA6bdfwQfe8EpPdSS5WEvr6YJyA07rmu5fWdBRhjygIKY3nAgYAvd4c/FyyJGPA78+X1vnnYefaOC2hhKqvVMjwiwy6Er3SEC8FFRJJJIUEEEkw1YcW+xKNFl9kzKaMjssGALTwfVTIiMBorHGUawSjGWyjCDMM0I2CgKMZDClEQQwIwYGIYw3wsyDcmjG4+FlmePbDj9TIFopeSIF57QpLFAwQPstMwnVE+UYkJPEKSYRBehTpAg4Fwg1aZGYhzCAfA/9nWMMwMAMcArlGFBookK0F4w4AOEQLPRgZVuwlhGLDWyk0kYt7BeATqJvKK3RggK5Z5haBQCJLmPEMh/yhheJABhDUGJFWrKEgc0BFQyZBQIH0YYpyJMglDlGQbmzDANdgIhULc40+LOEOWKTX9m5xMPaEhRtkQJce0QQOHCARcFGYIkRwcAyKEFIgnSzIE6Z4DRQccpEf1IIVEhdCP+ENUqzoBREu9g6sQQURwaCFASaRJnBEoBSAEwcQKgiRYKgAeA/phTAJYgqC9MAZBInCJmE5GStiz36PwlsoQuGLQeQIGM6jCjOqJhA+DAQbkeBDM67BCQO0giDEIGQk+ij/EGRIEBbO2OFGwKEJPgQDHKGIBCMaCI5M8IEY3wjFJOo5EGYwwgCMWKABbmGASwzkENU0QC8wSsckXkISjLgF4UxRCPMJ5BmaKGkvItHRa3RECwBYgzXEUQp3SoQRZDBlSXG6BpuGog8FQUQZuCmZRj4BkpGs0C23yAtYHAwKY9GpAbAZVIEcAhqCeAIfrvEHARoAHHfowk6ZIYeBsGKZBuAEDRISjbra9a51VaRBUNELVrygEOHjhBYMUIpgmIIGfHBGKCJQyo4WgRniuEYYRBrSXwgEBWc9hiYMgIJpCoQXOBjpN5ZQCgOEwrIYYCK6vjEQVMwBicRIEEeY0QMA/6BgDpko6UO+QJFu0JQgzCDjbTdbkDFCk6mBQUYYZhlJCgUCeXi7xC14EQMAjCBLYblFSA0QgSegUiByQIFNGYFUA6ThBa90RhoE0osIEEMg0DDAdgvCifra9771na9BAtc2gcAPhx49YinKwFpJcECj31jKHcQBv8a+wrt8AAfm0CcQYkTAjAIhQTeQygkOqLEMSKAjZgfS1Y3EkQYByOkrI9IDIUIEFY0lyDFQDAA3cKMgvMAAcgvDiieQobm5sdEtNdEKX7wBAAJQhli6EYa7fvcaI8VBeQfii45OLxdkwKEOOCMQXDSxIxwtwwoF4lKOfmEpBHFGBBIEjmCsYf+kwhTECzR3BwMIMxIkYK1AejBmgRDTAPCLwmQF8gJBFMQKsrDGNfRckG3g9RW4wOuNDwILMlzDF2TMwaQL0oUlJEQFFJFgcS3dixwYQAd6FQijd1yXRiIBqpHUTSdu2QlSGOAUGKrDWCJxhz74ug8o4ABB4uvZdhrAQpnABROr7FGBCGKuIFHBbzFqgFKCgwR/Hgi65iAJx2GDIC84pQGqOhAc1PmzBugEQdYQgEO20aUL9PJAGOG6CCACIbnA7x3+gN8qGyQZOGDiNwoBgIsahAQcWLFBXAqRIBYkfAI39LSrzerkhoF4QJ7SLVvFC71d6X/WmbJArMDAgdzTpgT/6YIHDlJPywqkCG5AyDeSQfOa25zmzDyI9EYqkCWoALYGcHlB4muQ92YQo5OuMjHUmAkD+JvQISYsBiaNitQWxBmtsAIAYpwRYyVTHHewwnGfkXOD6Lchl+CnQJ6Gx8FWfDA9/nHGxXRLmd3CF2IIgFvCOBU5RKMgd9zhHIDQkL8boN0CwUYE9Kjuq7Pi8ZCP/ONhoZDVvtTKBtCEhwsyCWETJJEG0KMasSFuPvTAAMTgaD1RLtKNCuQOhBfIGjijR1r0oIJIfHpG7mBwA2xNFryNSBSafRHeFwQWwRcILF7wdrlcIxCvzrgk8UbkX4iiL5bgu796P5BAtF4g5jZI/ytAPRBkzEEcGDaABJHBjLNjJKhqnMMS1NgFkhfkvax/xhoOCQvPYxSbApEDkmAAiHBIxBAAGtUNQCByA9FKCWJwRYByzDBHHHELQMBEW3MHDPcQ2hER0mQQFrhqGihIbtd8ZaFcGDd3f3JLFOMLt1ACAGADYsEKYRAAQCBvLIQIoNYFxGQNAFBsJMYIzkAMkzAJrCUOX3AJFpULKkALhaB2GoECZDAJzFAIX/BtA5FtBIEIbcAMyMAJhYBy3KADodAMmoCDBqAFRHZPAmFDzNALUaAJmvNexxAMfTAHYXALrCBSGdUMwiRKGsEIXXAM4uAMh8AICfQQskCBVMYId/+EAocQCVgoEIigW76HCF9QSoaIiATxBU1ngmQRd9J3COEkTpzQC71wBQEgAeogFjV3DMxkDcRQc+8FDq9gib7nC6VwC6x3VrhgCn/3DKagURwRPr1wDKVADMAjC5t2f6ZAC21EEN3wCqVAjAKxDaYgdP1kCq8wiRVVCrLgO6yQC4QTDa1gCr6AixmRDLwWCYSYiA7RQGo3hDYXDIyWFgnBjnfgjgdhjaBoFdeACE81iiNTCnWXCXeHCQLQHdrXIJY3F78gUCUhCBO3EKzAdQ/BCiX2j1aBgoIwijBTd0PzC7JgAQDQBOnUIWHwBMdFHTeWagkhbhABDktQdhwpFa3/4GOjGGT4MzW+kAtGwDzz0CLNMAcY8AIMCB64IFsLoVgTEQmldZNUEZADOYrHMmu3VDa9YGQRIADFAHIMAg7RYA1jCR+o4H70BYUNkQsVKZVP4ZE7qXEHiQu/cAoHcwZuuRJqaRDN+BB7mZdFkRVyt5PlUneQ8gq/gAsjcCWA2ZgMEpCZEZfTJzromIqr6CuOmZnykQxkkBqSaUt1p5W+0AgL6RqaeZrkIZiSSYogZZiZkAu/AAsTAABOgJq2CR7dgAiRKZkF6Zqy8Au5UF0UcJvEKR3MIGYfKZkhWXej4wu+wAZ6p2TFOZ2/kRVhsJq5ETSGOU69YH0GAAB+4EvU/zmelJGbuymZx2ItdbdFsHkL/NMC5BmfpXGcnomddOeatPALugQA7AKW8vmfgWGd2Jkdd2OYBeMLvyAIObIgANqgh4EZyDGgi2OY50JOdVkz4umgGmoWqJGc9hk6hsmev8ALIhAAF+CfG5qiYZEYA5octYOfv+ALUhAAE1ApKnqjZ1Ecx9GizgVdIskKltUIERAAzIKjRjoWBdKi2rOdFgoLGDIDKHqkUkoUtoEeSsoHPYk3HKefe9MBU/qlU6GjEcqjI+Sat/ALv7AGAOAAsxClYPqmp6EXM0KmY0Oh1fcLnWA6bgqnfNoS5mGlZDoJFHotpICguBACALACfbqoP/8RJTyaHStomKRwijFqMRjDqJg6E80gp4/qoiBKoQiJpn2wpgyaqabaEhECqDwqVYM6kr/ACn2RNqc6qynhqJ1KIj7KnKWAoL0gKBkQK7QarCLxJXP6qEu6nZT6C0zAnzYqrM7aEW3Sqckha4MqM3T5C4mQI5/wrNyqEToqItIql3aKmLFZMynZrejqEMQarlk0qNaCoL5gJX2ToelarwQRrezKB4I6qFqJpsszAcZAr/Zqr6jCrpBarXmDpoJDpAPbsAphK/XDrqUIqrCAprjwpPvgsBpbEPJisL3prqiAoL+wN0bTkBvbrQVrsIewnBQaCp3Qnb/ABm0xC/2wpyf/a6oQq7I8ibCviaaowB2nY7M3i6kda7BXybO/qZ+IugLAOrT2+i/TorJlSqHNiaZBsC7lILROy6cPE7FSay7Vyp1oamgOUKpb260fEzI6O5kh2rOxGatae7ZgCjZrOzA8m5/6yT8ZILfd2rV1y5oIi0to+gs5YQHjILB8+6Zp+7edYpBha6G/cC8CsK2JK6x0W7csS6Ei+gu3gKGIW7lHOjdeq7NAk6WGKV1o6gslYAAX0LSgm6mHQ0trm56Be6e/cAUGQCmve6qX+7f3Wa3YIrLo8kUZu7uMKrqMS6CBu6XAyRi1+bnGq6Gxm7zKwTiBG6pouh8HgF3Ry6fXQ73K//Gp1Wq7YtAW2XCu3Yuj8zO6dfui1dqCaHoKhwcJ6Ju+KgpC4IurgduvI8obcWu/1Pm94Hus1Yq9lhUAFiAP/wvAt7m+4Hs/gXuYg/udDqAM0MvA04m/Dzy17rqr8dsXj4DBR+pND9yuCMu/POcYInyjjfRIJcy2g5psgysDBrAB/rHCKUoM9Se71AuatUuuv/AGA7HAOAyYovjCE/u+6PjBRZyiTgVrD8yq+5usXRMAQnDBTdyYysVcL6y/ESzDaBoD6yIPWQygR/zCBFzAFYumAuEA2VDG8vl80ffCRYKVgVu1v4AJ3OEdcEyeKEjHIqSe+/uyaCoLNdPH5GkKOv8JyMpRpxHMRWjaCyfhpYg8nc7AB0WgGYGwyZzcyZ78yaAcyqL8yX8wB4wwTqicyqosCa2QC658tYZbycWJgsX6wj58xyGLpkKsI/Ury24pih6KxOJ7wpArCinmB73syxxJlVAVzFFMMhG8uRebqESszIsEl9NKx5l7t6lboiNgzaeZk4OZzSVcuhFMMECKpkSQu9RQzeD8PMwsJc7MuEd7zuMksuY0uSb7zhWHzUQyz74ryMurCbzwwRnyGPzslqqJKxsMtueMvblwMC3AD1ic0PCsm2PqJgCts/p6zgTTCoO7NxhT0Ra9O5xZnz/Tw5EawbUmsrjrAGbiziUdNwv/LTAbLbGtec6TWtCRuyw1O9PNZ54ZPS83La0fe86o+wuv0BdQsM9AHT/0WdT//LfbjLCuKsl8w71PvWMCGkJSraTa6dHw+wsWE8tbvWMQGmuzy8EnnKwyW7ZnvWMd2lxfbZ+O/NDX2glAK9NxjTMsCmR1/ZkF6tGZQK60gCHc0teLlNYZF9iEab1ivcS9eiXvoNiLNNfS59jSxwfDPMXd6QvLCrAkbdk4U6VxqdlA5r54bVl/gGSUS9rPI6aSidpR9VweXS+FjaakEKtODdtLk6SnLa1pjMsISkn949vPY9rYSdv2oxum29bd2Qsi/RnInTo6KgdK2qlsfb2wKaOR/zML1Z06wL3cj/q7hJ205hQBRRrehZOqncrcUNPRt50/uVw6CDPa7L0vtsqj8F03K23PJcS5eqvV+b00mwomwj2gSRzNkNwLZa0NfF3gzuLe7NrfbiLFt50J+ekLywPXEr40+53gvGnb860JQOoL2RoAavDhS7OuKmvhEzLc+2uhqPCkvc3i/IKvL37ayDLfInoLgoIlEY7jphLiBgvj2SHQHo26vBAENIqZRK4yX8K+R06YDn3bJq6f0CkAsTDkUY4pOl63MC7f8x28o9kXfnDjX94tKZu8Fn7LHr2ldpmoar7mzZKzA5xxC/7QZ/qC1uW6dt4vRUu98H3UWA7Suf9ABJfp5YF+J9EStXkeSVUdwS3YC2+QI7rA6I2uJniOxrEW1vMtmpiQa5su6CDDyLmBRfU834Bypqcwm0Kg6aXeJY+O6qleS0ru0SN5CyMQABSQD7P+Ln5r65pdmKyeP7uKC0agd+pQ58FuKYtr6+SsOPvK6lrJC2+wkK4g68/eJKTgOdJOI8UD57d9CbjQC46AIXng7N2OJ8Me7rcuMKqN5a/gC3YZAEaQzO1+KdEO79NeLIZ+2+8KCyUQACOQD9y+7y7Su/4u1ZMewVp5CzIQOa2o8MCCvP4e40QN6qyebLmgiklm8cAyvRmv8ZxCu8cOKLBg6dxhmiIvKwxf8vH/HiZ3Pd/NqZAAUAfs/vJagvEyb/JhMtjW/rKmINH4zfNAQvI/D/QTMqEpDyi58Ar80wFkjPSYIsBLP9VEwtlPDyi04HH9afWW4sBZL89EguGsPjq8sEsSAAxibykaXPZanxxefOzcCTmTm/BvHyBYL/dzL+PnzJ7mpOI/vfd2Mj9Q7Pcm79xdr+GhYON6b/jwEfeKr/Xbreus0AoHI4ORL/nkQcKVz9Dmbe2hAOQAcAAp5Plo0sKJH/oxTubH3iq4YDE1qvppssU87PoTso9d7/Un5OG2zyVnrPthUgi9T2R3kCPrHfxP4mqtT/x0XwjP7dG1RpoZwvxbgvvQf+GB/zBOTz+ppIDY/nD02C8fw7/9NCJk0x/Bl9AKiHoBVV/+QCLHz7/91HouKW/iVjKcnS//AGFA4ECCBQ0eRJhQ4UKGDR0mRBYmyiGKFS1exJhR40aOHT1i5MPJ1MhQI02eRHmSU6kgACQY8/dQ5kyaNW3exJlT506ePX3+BBpU6FCiRYGyekLm41KmTZ1u5JPpZMmUVU2G4gQGgIBPMY1+BRtW7FiyZc2eRZtW7cFrgpDceRpX7lyQk1BStVo1U58AAdSsBRxY8GDChQ0fRmwz4kS6jR0z5RMpJd68JzMlEgAASmLOnT1/Bh1adGGkSh+fRo2xD6OqlCub4jTJAYAW8v9G38adW/du3p+vIXoCN/Vw1H8ElWr9+uqlEAEurPvXW/p06tWtX2+4WBBx7o4D/XF9VfnITDAAWKCGXf169u3dC26VtPv8uYL6hBevPFMQAxJivQcwQAEHJPCh34KjL0GnCuGjk8rwQykrA7jip0ALL8Qww+m0U7DDpUJ6DcKr5ugLCw1PRDFFFQWjJQrTPIQRKqlCfO2PzGboZ0Udd+Sxx526QQSJOWIkMiM+7FJORFMYmS2DeqLzMUopp5SSGTKi2K5ILSmKbDxTRNQkBAMOSIdKM89EE8MWw9iyzT4O8fJLq0JhwYAJjEkzTz33tA5IIdvc0jjkvMSPJQMciMX/Kz4XZbRRzpgpA0tAtfxOycmqkiKCAABxtFNPP01rzUm1tM/SS1FiIwIATFQUVFdfhfVHSf4clUgGRYqTJJQYycwJfGIFNlhhG2ImDUlrJZIPTXLNbyROHDDggl+HpbZaYVv8Atki+biE2WZL2cAACgxo1Vpzz120m1mH1DZGPiTxttkS+hsnR3TvxRfNYo9t10M+WItXV1OCCMABYPJFOGEfcYki235h/APOgHW9oi9VylU4Y40J7GaSJdh9uMM/AiFlYpLeyKwOjDdmuWX19g0kZA8FAc/kkRIRyAl7Xea5Z+oYdlhmBQXhw1TlMJHAABl29rlpp0VTdwk5hPYX/1eTSQm3BHqe5rrrR9N4ImaqFVTWZlNKmVeglb1mu+2yetEi6LHpO9JsU0YYiGm39+bbqI6RcGNusiUz24AABCgmn74XZzyoZowVW/D5VrPb8ABsqbBxzTe/CW65JecuYrsrDsCStTlHPXWC/g4c9PlGLtlmMQwAoA7Vb8edIGjaiCJy14mj2ejX2AgAgDMyzz15zuHu4vf5iHbQZhsB8FV56zXnBHDn5wPR5kNmq8G268ffGxo5nnhze+6iMptJADogl3z5vYZbC/W7q9tmSUIAgALo5geg07K3hvutj3Amm4QIDGCBcZwugA/Ml/nQV0DiUM5ml6jTnRwIQQ6aq/9+FCSOoExWivL05z8dRCHCSvEEAoIwNZUa4X4O9YkU1hBdEkyfC09TKpOFIhNSmBAgoGRDIgrrgzpEDYOiF7BQaAIME/pLEaUIrFIsoYVIPE33mKiJO5Roil90FTTmMEEsZnFGTOTEIIoHhXqA0Y2OCoYW7FfGLCIpYJ3gRCNUNYP4vdGPelphGuiYxQPGixNYycwK+vhHRlJJjGQcZGPeZLJDoqJJjcQkleKohUJE0jEiDNhKWiEBADxng5lE5YUC6Unv1CyUpqBFc8Z1ylTWMkDXuAMkWSkXHgZME6i4hQj6R40h2tKYFiKGHDu5y7ncamK/xAULAKDBY1azQKT/eIIgmUmXsgUsE6zIRQymeUJrlvM9uNTlNp/CrYldohW9YAIAEIU8c9YTO8lkjDrl8q52vqIXUtjKJ6ZlT4JWBxXZ1Odc/hUwEsKiF2A4HCD2UVCKSqcZfFhCDhPqFNHFi4S06AXx/KK3ipZUNL2wQhRGFgiWttSlL4VpTGU6U5rW1KY3xSlL+RCSUPTUpz8F6k85IYlX4OIPfYHCPEy61NGwwkUb5aUrmdVEXPiiETciKVO1aphr9GEJd/AdVJdCNOHdRRNVzUQiJ7pVth4GnxQJq1g9osVcYSUXvgjFbFqwyLb2FTBO1eYh4ipXqCzLW3jkhS9QQcoL8NWvjzVL/1e/apHBErYuhgxFL3zBig28r0yQBa1ZImKFZVakspatSJe8dUjNvkKYDCxmaGVrlNJk5LSotSCzVqJZWJTgPNSg5WyFi5OuvkUjtyVsR3VrCl/4QhbinEA26Dlc6vZktKXFCHLFCjtv/dIXv6CFDFwCjOlW17w4qS1HtLvR4HWXFd+lhRG2Qt7z1pe4gTBuR9arT+h565vfvYUQIiAAVwTXvgcmyGKWsl910tVLl3gFgJugqVEg2MIPSe9HGMxM9jELwr/4xS3eUDxPGPjC9W1LfheM2tTa0UuluAQsQHwLLgAAAITI6ol1rGCnbNiTqo3TR2f8BhvjQcdHNkh8Xv/UFB8PMreE0sQtQIwLMWQmDyZGsnAPJJynNLmMoIQyLqY8YgDkobxZRjCH5OJlJMIwTk3MxZTF0Bczo/nIsHgqXdjswl4SihNx/gWVBRCAOtv5wr9R8Vz2TEEl5gqxIM4FGzJzPENfWM2NWXQBHfwa1kJaDJpSWaUt3KIlY1qsHfZSp3+RizkHoA5nFvVw/QSyx2R6e/lLdSl6AWJeiAEArh5orM1rJX6dxta/A/J4RPLdX/TC1wHAQj+wLGytioo4xwadcsfjXRD3Yg1IpfZ5Z90dbAsOzPp5b7efHe1wVxdSxR5Oucfm5vH8t9tagba02z1ca89H3kLrs3IgzOz/h/462tPed0HVRSv6/DtkznzwK0DcbHybKOGz3VeWEuTwh3VTOQ2duC8q7tiL9xVbMOJ4u9g5HlJkghYhr3gbSw7ZhdNaQSlHFj/H00Qpg9gXYdiKxWf+2IwTCeejWujOzxpygAoACwgfujWBRqlIavtBmgD0L3zR9L9APeq27NjHtnT0NnE3SX9m+lbUEOyvMxVmbSI7qaRamUf7nAlQhHXbKTp1QMWdSGRVdmZDHk8BqMHres9k2Kc2Kb+7y2qVUbXWWyIAQOQd8fV8+6ga7y/DclrXIecP5S1/+XJ6TlubJ5uL87Js0B+q8qQvKevahXq6FTIv3PY5fxzAKdhX/7QZvKO9R4LPnSfnxd65d31se29O5oVs+MOxulUy0Qpmaz0GMzz88r8Y9tY97PnFIZnAIxxyGETAAZ/IsfZtubveCe37j2nvaz5MfvN3Rf31bD7V3t+Y/lYG5OTHvvszp+zpPve7n01DiZZ7OQBElPQTQEzCIcHZP4U6ozmJsonTOhg4lFhwwAdkpCOSwFtTvclYuonrBRYIAP/IPg9EoQFynQlcJ9ubDLQzwXmRAPpiwVqKwBf8neKbwaxrNhvEwRxEJRDkwWw7DrrjBF7AQF6wQWUYPSL8ohW6ot+BwQWbu5SIvF/gBdqRAGmIQiksokfSKCuUnIDTQsGbOFywgP/+kC4xxCQj3J4r7AiIs4rdwkBZmAA7yQblg0MwCgUrAiE65AgENIllqb5fOAUJCIBwWcE/vB4ydCFC1AhUqwrcAzFHgBa8gcQ/2qQ2Gxtc04t0mzjMUJp3eMROTJ5VAkWhSbaUmL5EJDMh4IdUVMXbkUQsokSLiL6TmL+JqxjqCcNbfKBNwi4d2kWKODeU+EWtI4K+CDVinCJWpKNkpDeUgDEZmzgDuD4BgAQ/lMYOQqcy1MWHQUOTUEAMzIU6cYACC8ciSiZOYiVKtMO7yISeA7FYspNYAMd3fKCDCixPokSPMysxmzhWICX0GEZ/tJ5r4IN0iiRCXLkfxMBOyIz/A0gPhkyht1InOtQ5iizFzMiAz9LIDgLIhLrCpEuJTugEJpy4LqINVCxJDnLIjNooGOxF2FBDrXuiAOAjW5xJt+HImxwVs4uQneyFlgiAG1jIoFQdwJKr/Yu/CPk8SNuBgokipwQgyeIyqHo/wEsJRJw4WRAmCdCDptTKzSGGLsinqAQUBMTExTqPWEDLtGwcqESt7yPIQyTFX3iW/jGGurTLvukqBGGx5xNFk4hFnxsEDCilcRDMwdyb62IxuNKSVyQPieu2OxBJySSfDKvM4PNBU/jFhyqeFXgSz1Se4urK0CSS6MvGNWSCEmE71VQdyqxMyooRoxyJlsNHWoCB/8NRgw60TcYBzdy0zJnJQh/qOV94BWmSACEqTty5BvxqTeQUrJlpkIL0OVPorAkgp+lMHR7DzotoPAezq27DBAcIAAuACfF8Svkoz+zqEEv8EhrMhUaYjQ4YB/hMnS2bT9tKPZRgSZfEhTdQlQwgOf/sm0sLUN2svaPctRBLFdqoTQbtGyV70OOij+LrNF+gBdKBAqDE0HwB0A0V0O44NzzsBVggArUj0RK9FwdFUQi9trkTy154BfNwgAqTUcbBs1KrURtNjYDjtlw4BRFIwSH8UbdBtOscUtMijnrUhHS7BVGYgPOQrhhtUmpJhivRuCilz+Ggq1gEUU0EgBFIh//I7NKEITUxVS8yrcBMkDjnQpkAsIF74NI2DZZxg9ONkLePJE0Za1EhsLEm6Ec+7Zl3C9M/HdPHALLYRNIR2IpH2FNFjZV+c1QOfQzl8s1AcwRGlIBiYFNMPRc/3VRA7dQk/JJ7DLE3cMwRUIdSNVVrYdRU1a9WogqqAtFgFAJ5uNRa/RRNxVVOrY/7IIk/6wVWECcBeDVhdZqaK1bhowuIs6tewIQslQBXgFanKbppzVWFMixOaMlcgNVh6tamOTlwpdZ96had7IVbuAIbswGZTFeXkVZ2bdd1ghd43dGt2IxEvVd8+VZ9DVenSLrdCgX+kYBPWKuBZRm+M9h9ZYr/jvolXhAE2qEAY9iHYIXYPAk7m5tYVW0K7oImrQCAkSTOj7VVyBnZFWOK9qpSWDAPAdgMlt2YXIibl2UKBgO86SMFhs1KnE0Y2eNZmP0IEKHTQ8gMCpiFlSVaYcm8o9WwD1kWd0LZDJgGqI1aYDE9qu3ZpJ0EGEOFHdgKPupahDFasEVaqIiEluMEoV3QtKUW9ktGsUKu1fCh6aEAUOBaunWV/GNbJuOIiGkiKSieDNCGvwVcT1nbwQ1bjRiZoaoTAeAjgW3cV7FbyO0yjaAZTRAEUopOf8DczAUVweVcwsUIormENGjaWfBY090RF0zdzq0LgknZdIhd2VWRHazd/x4DiUJQIJulVd5FEzn83chNrTmAlug03mqh3eS1XS5B3GFi3OfVE9+VXuCliD7QQACwgXXYXezNEOTd3si9AzGxWfIVFio8X0U7hDVoXlcoXfbdk1x837gIBCKIAHHJSPt9FVyQo/xtprQRgtQEYFfJhCJIg5xy4AeG4AiW4Amm4JxaA4GIgDpQnAQGlUeCUgL2iD8wHA6OlU9sVBDmCPuYAwUCgBpQh/olYTNhxRNG4YsohD7gA0EAA2hxAEuA4RieknGsCBqu4UP4Az7og0m4hPJL0/8F4kaJR+wiYgImmn85pD9gxADgggt9YkBCqIuY4vM9YkHQhFIoBQPIlP+5vN4u3hEhXt0aPuIkpgpOYIQ6oY14KF421pAo9lwq3ilGsBpO4KLZcN7x1eP1OMmNCOPBHeNMiJ1WNZT3ceJDzhOHhEgwlt44loQl+hJNCAVM4J/1/WFKXpGhTOHarWJADktOkAWI6p9EIeU9SWSPWGSDxWFBcGQt1IRXQAVhCoAVIMlYRpOaJEdFZltN5mST6IRvyoU+cMwJGFphPhNT/ohaTlX74INDeLypWJZcgIWWKCU8keY0wcumsOY/xeFAuAQIKQVPvoVemAQPkCdWGecz4cq4OOchRWbI4wRY8IVb+ILiOYBPqMV6NpNkIi1eAlds1mZOm75e6IVOECb/AVgBbzDoMylnfC7WI1ZnS/EhVGBCXKjQCdAD8bloKSnMD47ZTQ0EJI6EZK4KUvBkMesFVDAPAEAAA1jjkyYQ3KSLfGaxKm7obeMEWvguXBgEQu46npaS46yPKOXodU61Xf4uHX3RcN5ppr4l60QNoPZKl4bpvPjoXftnTOis9c1qrX4P8tyhAL3hbN7mECljQOsFWZizVx5ltR4Qp3YMr2amqC6rs9EETriF7/pnUfCtAHCCd9DrHqnOROvq3CSaPnjp1aLqbpOFN5iNCXDYxuYRth4Ovy6jtz6EMj4sd5rQf4YF8WrhdEhrz14Pvo5suTriP5iEwL4Kqcg6XpCF/1CVJ0jQN9hWkRN9Hqia7Mr2qJkOuVsIMBvrgGwQbh2hUe4QbfUhbdMOpaKuvlygBUeYgMMptOhOEQ3tkOr+ndq+bUp66OWWhRqwMQsAQ/FOEeIeGlZK5bieqkxAhQmFNFpIBFISAC54WPnWkOmmD/Ommrcm40GZGJkOBYP0uSul1PeBbgJHkTf9uy/jA9vGbasYbKNWx1cIxgnwg7y2cOxANJEtbx26b7uBjcsOOVoYBFIKgBbQ3RPXkC+FNxhB8FpRcOw2m2XebwxcNVOIAVWhAA40ZBxHDAzfkh4HlDhObxf3oVIAQq2za1VxACgoaCa/EFR9cufBZlV2ccHuhP/CJvJcSITmSFnI9HIMudVJgXIY+XEGd3FB9mci94VWuGoLkM43vxBibZM5VxAp73DlWOZ3IvJAY4PZcICfBHQLAXM5p5oWL/PcNoUrbzZNEKZSetolj/TBiHNkIfQQymEgv/R2fvBF/wVZmM1p6jpQD/XAEPRRKfXTMPRLjxDtXvReEITZoGhtkPVZB4x87Zdb/+k/xu8y5wQ65W8MNAVpAoCBNnFi542CPfaizOFc1vWp0O9nX8M0lgAsWAdrJ5B1lRlkL1kk3uRuN6tQ0HQQ259fQ4DANHcBCdmxUXdaVnZ3X2U0X/RXKAKN1YMBv3f3mFqAi5Fb5nZ/NwlBHr//Xl+DzHCAG5jbg8cOidV3kWH3sHb3ZWYFcJ84pDGcC3hajAcQxQOdfR9iHB5qh+fmTGf1QMNdPwdWlH+PhJ8blk9nqYb5k1B1fNRzSTsUPspjnPcMnf0cwUH2ff75Xc9zVheFzgpng0d67Hjcld8hl192mG92RWf1XJABVZkA4H7tq/eMx2m/7YFywH76ydBvl2T1gRCAxR52tCeLr1Uf825pyvb4n3dwCF90R6AdCj96vEeMv1m8+/Fr0u76p//wRMTAU7i+/hiFu0d8sXCGNVj7AgJqt3/7qoB4yZ+48OpfgdiazF8P1PV8DQPr0J+Tb5/5XLgCATAcIdBp1VcP/+5DojB2fNi3QCufeV4AA9sPABlQhovX/d3Y3N6XXD7oaOCvinYmbNJvNjHAgJJn0uWvDtbnMzB+fekXfRgn8pDCgOKhAPTjfvWIXizKkuu2c/FX5ksIeVb3hT5Imv4BBHmo9vUHCAMCBxIsaPAgwoQKC0KT86TPoYgSJ1KsaPEixowaN0b884fPn0mhTJEsafIkypQqV54MlclUrl8yZ/7q1UcCAAAU1Lz7t/An0KBChxItavQo0qRKlzJt6vQp1KhSp1I92EuLFo5at3LtSlEQnzuROLEsa/bsyVKaQuGiObPXoBAGAFjAIo9f1bx69/Lt6/cv4MCCB/vlhGSN1//EihcX6sPnkKZOaCdTTsmJEy1fbmtKEpFzAhYD/giTLm36NOrUqlezTtjw4eLYsjV+DDnS1O3Kus9yytSq12Zek1h8vlGuX+vkypczb+78ed+rWWdTrw6WDyOyJnPv7o7SJSrgbnNxgpFTwgzRo6Gzb+/+Pfz4g0stQVz9vuLGfARpKpWSu3fekbJWW+OVFwEA6C2DnHwNOvgghBE+CM0csOF34Va1icQSgAHqpglmms3kCy6lMJGTAyvs0s96Err4IowxyrhXMFhheKNG12V3VoceosWJJq+IKJMvt5jCBIICtDALPy3O+CSUUUo5pQGlPJEGjllSpB9//qHVo4//LHWSSXg0FcnKFUkiMItoVLr5JpxxOkehhVpm+VEfG1YGZpjfZVJKTG/R0kqaAASQASgsyrkoo406yleN09l5oyCO7dgdn32SpFYntwzZCy2yiCGAoRl80qSTj6q6KqutJmQllpNeyGV/HmbaJ5CwDMnLLbeI4YChF5zqKrHFGuvoNXfUKSt1eOrp460ejvnbTLnQggsXGAQQwAWxMHgsuOGKGyMxWBXCLHU6aqdptJi+FOgv1vLCBbAAjPCJouPquy+/7cGK7mwfdanpdgRvSiCRuNCSCxc4AdCBK/1KPDHFqiW7LMBdOduudxxPdllmvxRJCyxCkGqvLqlWvDLL/y1PVa4W52bclboGq+SxWUC+AtzImMQQgKEd6JKvy0UbfbRRqFw5c1eOCZIJKTavhPPUZAIH6itvbJCTAEJI44/KSIs99tjX8LEERExzhKckkknNoa1rxQSqKVdg8BkU6pC9N99jlxuF2hvV/LZZVJ8Eoqe80NKIeYYe4EebfUs+ecVKxxr4RU5DTfiX3ensSy6tsOGwAy3EUg/lqauur9loY24R225z3nllY7LSSy6dEIEgXVAYA/bqwQvv6t+vf+XYIevOPhnOoWgC0y2CEBesGgbgNTz22TPKShSXG+9YIJcYvjxuaKkVyi2sgFGvBCucrj388VN5TR9L3GH8If94RiI7+bq1e9krMhGEuejkBgb4lvwSqEAXEaMLVpBZ4Cr1GOX1bze36o0p+OAZAAjgAnpQzwJDKMIGseIJZHjdR8I3vgq2pGqTkMLdADCBFbApbCO8IQ6TQ78n3E9tgeBDH/bHwj71aEB3gMFcAkABLEwDgTl8IhRVg4wwPJBp+kneEAnWoRdKYC4CyMAHIxfFMZKRMCU84cxSKL4sGow7goCBAOYygRnMwh8+KSMe89iXawQCCT1E1w+DyD82sssUpRDDBgQSgA6oYR131CMkIymVKQIOXVekICEJlggZOEAgpQOGJEMpSqecEV1qXGEmy9IJI1ggAnOxACHkYcP/UdKylgm5xh/sJyuwCDKVBlPLFTYQgQAYIABGKMYjbanMZR6EkpO6Yq186aNSdEITkwCDCIhZzA5QQozM/OY3S6klDaFSmrjpjSbSOQcWxNFQFMiD3sApz29eAxE8zBIvx2JOC6YznaVoBSsCAYNhcvAAdRiHAf6RzHkyVJSUFMSNoOmlfZqFmv0kyytwkQtZFGKgXDsAFKjRpIaStJatMOGNyEnRsoQCSOnsBCoW1gtf8OIVfGAB7wRwgBvMAh8l/eko63lP/Axupd9x6VpacYtciIhErTgi0ArKUxACtaqQfOh9aDVRo1LTpRjFBS+GRNP08QEGUdXpVIFn1bXm/5EWUUBjs4D4LIq2tJ8wvZZ4aNILhbHiD2ZF0QVuYAxvsrWwUawnEuaQLj5gB5OprGs6Q6FUpm7GAKCjBS1eMYi/AsABF8DCLvChVsOSForJIEMUIBobrUqzq/00xSuWOqTN9KJXt6DFGxrX2c8aYB8LLS1wb+hWuCpmY4+tZjo5gQpZ4CKvm/mFZXNx21zg4g0WyMlu1UCNf4w2uN4VYTcQkdjFSPBSWYRsJiQr2+fSVmG3CIYnmlAC7HrWD9qY5XfzGz9moFa1XhFYNCtY10w8L7aUZa9bLksLXsCCCzKQAO8k0AJApIOw+r2w/NwahsQYl3wW1UQmlAuL5iKYvf+17ZUp3hCDen2mCa7wLYZjvMBuSGK8XCkq4VqaiR2HghULm22JzYSLW7yCE2yIQQw/UwNCoON6Mn6y/JiRhtRyBcBbbSOIQVQKAwM5yDThFSsQAQYWRNVQFmgBIcaBXyizeXVu/UKG5FpOlViUwCImsZcRTKJS3CEIISizACiwAj8so82Glh+NbZwjxpo3TDqOrI9z4dw8bwZ0oXBDDDiA3QQdYAbVo+qhQz08KVNZI5qLGrT62Z/JdpnSbsHFJcZ8ss9k4AZhXLOoc903XEQBzhmJXYA+DCJUjHimrg4yLQbBBBEIJCcBsAACsBCLaYhW19YeXjcmsQTFXgTHlIH/bGRiKulje/kVbAhCFwkY6BWoYbDWc/K14506Ugcic/vZ3LdVrd4DkxvBuMDDFUogAGKe5wJQ+EQ1RirvhQeP176mCLDPIuyvhrXf7MWFKf5ghBhsII4DCcAEWiBtgSic4SanHI23fbwJFg6pdx23xYnUCwZrog9fgIEIMGCQBBnACZZAKNFOLvTJ0Xsip5waUvfd6hL7AnfSTZ8mAlGGILBA5wSkLwUy4AQ/uKLCCf3t0MO+t6s8XH+DNGQnvArb9TK9F7gb8m171QpOIEIOTKj6prErgayvAAuAcDd3cS32wRst20hwwyEkiMWShAK5w8brc5tO3bjbVra8aEUj/8QQhJzn3VAT4PsNAAEKgVSb8KZfXTOm7BE+qPCc6Uwvq0U286dX/hbNDQaJPEGJPLxBCBxH0LawC/IDIGAGWNBDKgxgjnfjRfCnf77ReGGFLgAxEiAyQDpN0QpYvKIVarlEJMByhzaUAQxMCEIMWCCCDUhAW53ftAAmcIAWQEENnzBANUjOD34oFPr+Dx4RvIABwAAMsID6iYAIhED7Wd3VvZ8DOoAETIAFjEAH1EAT1MEoGEM6oM7+jVZ3/R8ITs6o7FznCYADTMDnUcABXAACIEACzMAMQAEWqAEgfEIsKINB2IMOGsA8zEMI/iD8+IQQNIETQAEU1IEa+EENxiECmwhE/i1fQfgDi9iRQvVfQgEhFt4QqkSh82WhF0JRQAAAIfkEBWQABgAsQgAZAKEC/AEACP8ADQgcSLCgwYMIEypcyLChw4cQI0qcSLGixYsYM2rcyLGjx48gQ4ocSXKgv5IoU6pcybKly5cwY8qcSbNmQwk2c+rcybOnz59AgwptyURgtn9DkypdyrSp06dQhyJ7IvDRyahYs2rdyrWr16CtgAjscO+r2bNo06pdq/ZPDhYADKibx7au3bt48+oVmUwLEC0BBO7bS7iw4cOI1YZFUkZgDX6JI0ueTLnyTLdf0uwI4EAdZMugQ4seTVphXyBp1nwRaOlz6dewY8u++woIkjVryDh2Pbu379/AkwrS8aVNboES1CENzry58+cp+/ZI06YNmTIRAIyCzr279+8QaRX/uV2dzBycNvwtB8++vXvmgnJoqW59TpEAEtK938+/v2jp1FVXhhxpCKDdev4lqOCCdt0y3hr0pdYHTkIwaOGFGHIV33wRpvEHDAZI8E4/GZZo4ok+OfPFdPQZl0YgjQXgCoko1mjjjSs5WESLxpHxxx04NcEbjkQWaaRFG/KYWx99gDjBO1cdKeWUVBqkIos8ksFHIAYEEIAuUVYp5pg46shjeXcUwgdOToRJ5ptwXsiIfGfWdwgfTtoT5558+udMGD2UUWcactwJhpe6INjnoow+5yAQbgy6xiF/TGgAF4o2qummskVC56BpCFJIkwZYoA+nqKZK2p+B1umiIHeK/9GlMqrWaqtkuBQBqau5/XFnIQ4YcMaQtxZrLFuX9MChq2T0QakgMABgAT7HVmvtWX/qQIar5fFxiCB9iAFAALRea+65UeW6K7dk3HHInYk4AMAZmaJr7709XaJDFJGyO8e7fERSggEj4GvwwTpluy23bVD3rq9XGCBANm4ibPHFKPGCBBAMV5daId/+kYiBedSL8ckob5Qsvx2nESrAmYhgQAcmp2zzzRBZU4a2Hb8K8CRSANAZsTgXbTRCvSBRBIQd9/owIoMYaAmNR+slzkCk4EIMLLxUHdvK/Xbc7LuC/MHJBgbY4PVe4lwtEBIReHBHN2uTZk0aPPdsnbcAc/8ShAETxEN03Wm1PdAlhJeW9NJ6W+fuzwZEIOPgiZvVttsGXAJONJWD1kkPLOtt3ruH9MGIJmxW3Lnllw+0Biq3pJHM6pHd3cPCejtM9h+hgGjBParT3tXlmDMj0CtLCH9Y0kAwnfsaIAPciawBAFOz8lkZbrgB3TgjEDEBYIP9Xp+HrvcaL8MsSbD0YsW5udsPBM4k3ywUv0DcvELQ/QY4Ywrx4mBEEQQSjAjQbXx3uVveGpebQJDOdJ0YWAeo9hRmcGAy4BAKOK6BuYFA430NaR1B+JALg1yjIB00gDhQcYyCWOOEBOFEKLZ3i2AIhBNhQOBdmOe88/mIdIIIRCj/IjaxrORwdokJBQaAAo5gaOEPIryFFhghiTlYI4QdFMcrDtHBZzDiBfvD3OUKQTxc/CETkSBD1wZChmO0LhSm4MQarqhDtoQCCFFoXIu0RLo7hSJqAHjE9YaCCzCuETGskEP9WNIMGC6kFGm4BAb4oMK2oUIHnGtGERjBkPuJYxs94KBAkLEGPkSBBGEcSNuYEYpKBoOTKuQGElookGMgARzb6wY36sgWa6xBB2HQI33a1Uc+aOJsBqhB8JTyjTJwwgCsmNLlMNAHgYjjGBjwhUCSQYJn2i+FkugD/+aASlWq8mqToKM43NALa4pDE5EgiBVuwUu88FCY5fkX6QJG/wrkqCMrk8gFLDInzbZR05pkUEFE+GcAHPBinOWs5DnFgQhVdiENGVRhJDIRwxzWsy53zCM+2zCgPgZCEKXog4FmhBVn3GEg1RRINzTBCGt0oxQGoOdAmoE4ThxQINCYhAF6sQ1oeEQcpihEMrQoiVBkUBykKAQzwNEKSaCCINd45iTeZ8Or3pAWtTTADAvyDVRMQhLEyCgsGHFIa7Sig8gQaigWuRGDVlMc4MDAF8DRi1vQ8ZsFYYYBIXrO/fX1am3jBACiYDxnIOGvBiCGBzL6UbT4sgfBHGlq+qimUHA0AELKyhyaYQDSNkYgkmDGH7rQB2dMYokaZSw4iEFJgf/gogctjIRYEALA3jJ0IK+4BSpyEAltRiINBmhFLkoBBEEgoxUcQIZAUAGEtDpDN8TgaGQF4gEVMoOTKijhQIgBhFYY4BpF0F8r6MmBRV71p7QgA92C8YeO2LVtxwCAGwpxi15EwZsK4V9YcJlCcroThYcgcNvAoQUAYKAQazAeQcARASRW1izEeELzRtqwNEQPYJpoJeDkscyk9EITAwlAFKwpkDtwwHt3OO0fUPC+a+gmshhoZ//EipBI+PjHQPZxJxQCjlbS8hcGkPAziSGOTERhl6TAgHQNwGAD8GG2BrChAUyBBAP84RvPCIBODZAMDph3ICoAByVLgYFFrgH/CJgD40BWY19qti0XAHiBhJsRAf0FOIWRWDFhJUqQbkgCgFSOQnaiQNqCkGDMF+5KKHrwBA77bJ+RMIUYvAQMCjoFHGT4hgivZg16AuGlBCGGATgpDmSkQdUG0NVAdExZjURzDV0WyEChaYAwrJgg1uBAY8TRjDvIgteMSPNA5sBrTXDgp0ojCIoN0LUudGEgNIjpQNIQDHAww5FhJN4tftFb3orDzuLgBQByLRAc9ECEAmnFLAkN0xwOOoUGYCEArdGFXCQjCgAgAayxHc1Ic8WXOcjsSJ1Guj8UwhSSEMgZSpwUTqQGN2sgQUQNcEU/D6S+kdDEJEzxPlXHUyDJ/w7JCw4xkIirWhwoOPlATGGAO3BiEqGQsEB0oG1fnNkAQJDDQJA8bYHIAQAnPKFXvQfpO8SFA0M+CC2CvIY5BFm8KLyvOIgBAOQOBAkA2Mb91gCAq/LvDrohLP/EwUV3iiMKY2Yzu3dOc4NrhRhR2DCHl8TZPoSCEwMbwWBaiuqBrJiu+gOhQL5wQYMMWccGWILXDeJb3yrEeFg/Zf2kC3mCeM8gnB9IJGCoal+4DcWdzwGc8x0B8UGz9QUhhiaAEAALZ8RwBzUANyIweQM8AQDRSOE1cPHUFBZCC/QWiIH5x8qCIIMG+4OFAQmCAo/bHSqTrrSl99bHO3GiFFcAwP/EBjmUOzR620keyB1y0JADIl2mGOinVwnCDEkE+f5RRwjND3hCWHICAz8lEJEAWwPxDRlkXovUDYLwcdCXDCXEUX9lQwXXBzqwbGKhP71gBc8gU0PFEbgXU+JQBFZAEEBAArVGeSnECV2mdgaRTgWRC7tlTXkVgBiAZNf3FNYgBwm3fY7TfXwgCaYwCAJBCOQnFLCwgAVRX9okED3AbAVhCnImEM0gB+AAVodjAMfwDCLGEcjlNoXQAwfkawbRTrskENjQBifUCt0lEJqgc0BQUZFwRb8AAJ8HDk/QewPRDSogdCenAu/zDDhAV7dnULUlDqFAAgmIAVzyZwXBCwr/xT9uQAIMhQj3cw0koHNbh3wD4QwBAG43qBR4p3eWNjp95CtxJAIBIANZgQt3gAFPoGVJVgoVuAbmdQ0BYH2GRwrgoIWFID7ggASo8A2c0Aok8EqfxxEvYAWlIIxIsIGidxBzEAjdcA2swAefdw0qAAvggAo/53umwAs4JRB84AbdwAxf8AeU1U7Q4Ax/EAZrQAytdAs+hWWwqBHHkFIAQAOhwAoG2AVXBg538AQBeBDwxj0YwAzEww1mpQIBwAioMGUCcVja0zavUARaxgtfcH7JFYOfqBTZx4MNUyh9BC6hEApEYADBkhXBgAu8cAsQ2Qy30F+3gAsGIIyCKD+v/3AJpqBz3GMKlzA7cTVwG4F5vCAJtFBroQBZBIELmUAKECkQ0cAJaFUQrnVsBNELl8AJPPk9TYUN3KAJqJBRyaAJmfAKA5kRyRCTvpALMVk/4FAKEKKLneRJYYAKxNMNMZkLfXULPIkIxSdCzoAIc3AHc1UQdxBxHZkU1nAHmAWSm+WDnGAK9WUAo1CE7IFTZ9kVvUBLKcE/SWNuBWFoc1kQ26ACx5iYQBGKAbJ9LgMrxTRylzABBiAElskeaXBL0FGQBlAGS4hCBsFC35RCZImaQlEKlAaSl9Zwh2AKnRADgYEhXsQBOiBzzsE/zxAF25B1BsFygEUQfXGCxKkTi//ZmCDJcEDEO5rmJQZAOfvRDcSQDMkglNWZQsRQeAdGEM2whaBZaGSgeOGpE6qJnNbhLJBpCvEiEOz5nxqBb1O1PyaUmQ5aENHgiQpaE2GhfcjJRz44coBHMBX6od6xmDlQHAJKTN3XB4hgCqFgBEJzFCD6os1xDHm3miBJiiP5B6QgmQYAAJYAoz4KHBcqoB7TBt0nPT4pAQFQIT+6pLKxfiQqoB9TpHygkyI2ASTGpFg6GslgBaghpB3mmn1kOqagaeLXaVl6ppaxGF7qIg7UfUFUCqbACMGSBwmKpnaKF066puZZTJ1wigHQAXV6p4KqFlvapWs6NlKaCabQTyH/4hmD+qh5oaZryn1FKqamoFIHAqmaWheYMak9WKSBEAhwegls4mmbeqpdcRo0KqQ2yll8AEe9Iy3Ag6q02hWL0UNCqjtSqgljCgbipwwUV6vCOhTD8aRr6jIfFqaZFoQGUgfD+qxPASCempwmhVJxhDaPAa3aqhS1QR6e2kBFegijAkel8Df5Eazbmq4yUazT2iO+Eq7GNKZs4CWuoK72yhPS2q4aKqXLegnBwgWBeq8CixLi4a3TaqJF6nBjGgoDQwGDN7AQ+xJJ0q52Eq7jOqZSwBkUE7EcuxLMoAVY0q4lFa7eN6aAZBUdm7Il4SAGO62PCa8jF0foobI0KxIT/0uxyEqypmgKpRADgBMP6FqzQusQV7Kq35o+RVo2cKRpXWKmQ/u0FmEmFFsde1qgptAI8tI+ULu1EjEcy0KxfEeyfBCzoSAzHRC0XJu2zbAiRnuwfBOuf5CiY3qSnYG2aQu1Uju1aEKyIbO0bGAgqnC3gqsQc/K1U9uqUhqZpoAJ8tImg/u4ngcogqK3HiOSYhuzpiAzJUAtkAu5jxI2evuy4WqppiAFKOminTu4nmK4U9uafFs2cCqZUvMPtZm6EMsqk0u51CqlfcqcwSIktWu7AqsuoKu3VeuDOjmmA2MBVyq8W5ssrKu3iEqypJuxDiANweu86aowurtHbxuub/9qsgEAAJCgvVCrMRzTvcP0OHz7qgsbLLRpvkMLNuqbT3wLM2NqCssrD/Jbs9xbvw1DpPcbMPlLROXSvymLvgDsMdBzvycVuyMTSAGLwM9KvwDsunzbWWNKCjjRAnZLwbSqMwtUv+jTpnx7B7w6pr4zDxMMwqi6OLjavcfrg8tKcwEgAMrQwi68qRa8wNOrs4KQv41gIISgwzv8qAqEOz78vRbrd2MajjaQvUcsqDC8wOt7v32TvwJhASw8xelaPsVbv4gLr0A4pgKBw168rUlsxREyKVissGM6xABQB1Kcxlh6T2zcYckKvk7MswIRxXb8rOWTx1SLtO2ruDQHAE//EsjCusaEDK5YPLb5KysCYACmysiPisd5PMPdt7NXO4SYTKshRcjDxMTgi548K5vZGsqa6kvARMp7i8UlO6Y+GzhGzMoVyjywbL+yLMljKi64vKkfucsji8Uomr+JYBLBLKiXpXCkLLqvi6MLu6Nqs8x3mmGi+MweJsuzrL8CMQ91bM2fOMy7jD5gOsDJm8gCIA3ifKY5uIO77K7cTLpvkB2WEM7tbHChGMOb/MPRHLtR5wS0m89LygnHGc/dws130rulIDMjwLkE7aNrgAOZ4TIWfdEYndEavdEcvdFhEI2hGtIiLdJ3EAklKWISkA16EtEvGqAIXbGyTLrzKiP4/8zSdfSR/JzHhKLQoRq7QywsNW3T4zOemZXTVgzNA7y0pYATqyzU/+nSxhHPGNzLKWwK0WLLTv2fxomhVFvOhjzANRx+OHzLWY09RH0mRq2+nAyq1lp3gfTBZT0+UN0iad29/pzBfcy4BtAEcB3XyhOkvELK+0rV+SszFJAPfv2JImqsaE3ICNvLNRwEGpvYNyijhsotdX24+sTNcJyeAWAL/BDUlH00gN00bKyrsnyxNGcgEyfao100eXo+R93ACm1zCyubMkDWr+01hdq2gX3BXz3AmOskgrPbHyWpepTZk1rCCk0pyzmmRFQMfW3cR9Op+KTcXrrWSYvK9QwAhP9A3bykqnsnxgSq0HyguJkgAQAQv+CNQLdqadhdoqYs3Au7vGXR3uNj3fBt1+zL2c9dCnSbDfiNPeLNg/HNg2N8v0r7y14SuAMuPN164L/dum7c3OdtsvKCKQ9OO+wqoBKuWdts4TGrCbi94auTrx4+tebc3KXDCPX9s9Nt4gYT4XoKtmRgwtxcNjkK4BKjDK4t49cyHF3wre2q3fDau/PKo0BeNyhe427L4neSvCgJAFwQ40t+Liz74Qx0sP09zy5uCiQ+M1Z+5UH+KUXuqQmu4KLKnE6SDmNO5sbysSF75mtazBYOq0Rww8AA50WT5ZSr5dyC1L2sqKYw0z3K5zf/c7PG66U5y+J9sKyAVOWInjJFq9ZCuuIs3tM+KZtnO+kok7cy7OE/xOIaDHgBgNWefjGKHurledeEbZIo6eOpbjFXkrsknKHzHcmZVgqykqmzfjCgDsCAfsVQ/gdBHIQSp9u/vimr+8gIvtnNfbGTgNvKvuyMgrukPOw7DeUl602B8+bWviefC8uALui9DIRRd70/Hu5kAr3xrOVT3dwKGwq9rgrrzu5V8r/lvHDBLcskqaPCgu/mQrwvLeFGLraR2QgCIQP3LvBS4u4vHdXC5Oq9PHKasLx04fDGou8FP/G5/sbLyQkyEAETA+4aLyUKHPH0Ed+P3dxKS++BYQsn/18sk9ADVqDydN04ad7L3/cGlYwH1T7zR8LxOK/cqM3ikpzMAFADDS/0KJLyOJ/zLUPbxZ6imWABAFAC9tD0Tl8iPRz1Xc0w8e7yvMMJPisQl9z1eyLCSgz2Yc8rL8LtfdMJLGoA0q32mlLFbq8kmD3q3D62ocAGOzo1eN8oX7/3b38mFB/TjFAKgvCvXF/4CuLIiM/3dTLYLA67Cp9MQS/5JaL3lW/5WdLlSN8JF18qXez5cQLGoV8nMbzzg272ISLgqg8nlN/6os/Acv8upjNEEgMmtf8mmoz7ud/o3B6qrTS+fpD2wW8kg0z8E47pcs8HnfD4VK4ezV8lCuTM0P9v+ZA8/ZrQCEyND5Gf/d4x/N3P9wffvpHACTJjAe9g/lQyyumP2ZQq9yfFCTCAH7Qv/wBhQOBAggUNHkSYUOFChg0dPoQY0aC1NTrCtMGYUeNGjh09fgQZUmSbNWTuHEKZUuVKloX4cDJiQIAuiTVt3sSZU+dOnj19/gQaVOhQokWNHkVI7AmQNSOdPoUadSOZOSytXj3ER5OYAAAs9UMaVuxYsmXNnkWbVu3aoqF6PJEaV+7cjGTWYMWbkk+kPw4AnAHLVvBgwoUNH0acWLFCijku0oUcOWSaNYHy4v1TKJEEAzUCLwYdWvRo0qVNo1XKVPJq1hnTpBEk6PLVQn3/Lm0wYMHead69ff8GHlyxW7itjUdeA/uQ7Nks+VwaYUCCus/CrV/Hnl379oLW5Dg+Hn5uyT8omTfXG4mIzGL/uL+HH1/+/LTEoqgWnx9qyT4pz6PP7AoBAPDEPfoORDBBBRdUiDj9HnyKDD5U+m82Qfp4Y8AzDGSwQw8/BNE3a+YAD0ITQTJppQov40MQv4zgJ0QZZ6SxRrXsw+9EHaeqSkX0shpkAgNGuMdGI49EMsmcHNyxSdfksGpFrP4QpATp1FEySy23zNKaO3p4zMkmKbtKyij7iAEAB6Thsk0332QQxzTEdPI1M810br0AbIGzTz//xK6Ut+h0Mjk88VSJ/48runoEUEcfhRQ0L8EktEny8EIUpT7W0NCfSD8FNVSz5Ky0STL6wzQvQdzwywYORYU1Vll3agWI4krVUcLLMp2Dsw70mTVYYYdl6I4cvsB1xxR3xYsP3CZIx1Nip6V21mSsAGLOZE+kqjk8+2ABAAmykbZac899tFYkttVxTvTujEEmmtClt943jUWWXRMpK+Tdq5gwIABIXrW3YINpvDZbffdVzl+WBAIgjxgPprhiD9Vd2MTkLPtxRU4NaIIfgi0muWTu/jg2Ywgv/XG5le5wwAAZ6jG5ZpuxS0YLhVV+8NSWzVMpEM4osGfim49GujR1m+JZP11/djklgSbAMv9pq69WDOV8m85vWajPE2jNkbEmu+yyct6Za/G6hRroQwwAIABdxja7bruHegUIJJhWOzwy2wZaCrgfoftuww/PSRAdtu77uNf6BXw5rgzYEHHLL78p5x60bfw4QyNHiQ2BhCgcc9NPHyjvvTsPrySOI+/DgQBiMBp1228PRIcuWG+djPJAL0RIEea5vXjUNeec99aeBh034583/ZYiVlfeOK8jtxJ67REXJActqj+ObdDhltuAcrdHP2lmtNgcfOPKgBL0Q4IQSJXq0sffZumpd3+1v0EXHAAIkT8C3qx73+sfax4nPzAEgHIFhCDJnPGF9iVwNZ8DHRgEYADSRdD/gwaTXhEsyBqWRU4OMavB+T64wmoxwnsjvKDPQDcHgYwgHypkYQ5n5Yww9KAMMFwN8wB3B85YAB6102ESYSU9ILgBiJK5Xtv4IAIDTCAbSsRirCLxwidGRnyA4wMLDOAAZZQui2d0Ew992MXIpCF+keNDvGZiRjTWMUtMdCIb6fI/MAIsAKOwYyD7dIkeIFCPc7FT5FxyBbj54X6ChCSSeKgDMhySLhhsmyD4AIYI/AWHkQQljXBRhCZacjy+i5wm2TAgJ3wylK/8ECENacq4yDCTfOgLAGqARFj2kkGTrCQt5SLEnwWiD9IxQAfw4UtmdogXSACCMOcSxZb94Q/S/wlACebBy2Z2Ez6EjEIepRmVL/7sD4G4TW7UQUdvthM4wBxnXNwFOCphwkrjYqc79XmaZ+Yonk/hF+D6UAhOREds+0TodcApzn86JZFt608n0iSAYiTUosCxRhko2dCobAyMjOiEnmzBzYuWFDS9QEIR+MbRkZTwZ3xgRChiEgBPmNSmpFkoS6Fiy5dGIhSLEuBNhboYa6RhozqN0EmkKIlQcAUAeBhqVA+DUn8iNSTURA8fJtHUrpyBpFIFa1k60YNwWtUp5fzRc0ixSgA0wZVhhStSmGHUYJpVJHz8kUs0QQo+bFAIX41rYIcyymy9xrCHRWxiFbtYxjbWsY+FbP9kD0sG3wXCspfFbGYx+4c7TIITsdOlYEUbllDYyq5OwWTHXlKKQfhqmaOFrVAoYpHkndYjrvvaS0zBCM6MQB5vjW1wa0LVptTWthtxKXqMGQpTREJIFIiHcKW7k9JGwTXH/Qgxm2NN5k4CNxZY53TFW5PZhqkNxsVuG7B6mXN2l4pUA+x45VuQ1Kz0vOnlEdSoxNxL3FMa8Z1vgJm0EfTaFn5QGygpTJEJGIxRGQAO8HgbY16NFNiueG1Of0qx4LABI58Rlm59P2Jhqy6wZTA1hSk0EYMAzBHELybIgD1CYp2mtjkoVjERHOgK4MJYut4pEUhozNHkzmYvKdaEEeL/pooP+1i0OLLvjLHLn5dKAslGcKAleuzk2Mp4MtjVLosmgWQBGaBRXA7wiII8kiHHc73NukSKOVHmPDQZzWGFclTaLE205kWvcp6c+e4sX0HdCip7pqUbW6ZJTsiZkQ8ctHgnRWGAXvguqm20KTghhg0CJtLTJZVcEH3Ih3qrD52Qs0wAwAU7f/qmqDDtHq1q47wsN9VdYbWrg+ullEFm1F0sMmb+wFxTdGJyTWi1ri8aal8jlafsDQSxQxGwtiZb2QmtlaGbzdIwY6UPgpC2qpF97dHiqzW/huGbrTLQDZsiFGx1K7kFmzB0f7mhffb2Idr97gG51dry9ibGHNfQ/3nemBEpdjdb/wrwuJrbb/8MqMERzm8ALJzhYKV3fuoNvlKzKBITZ6sNRHZxqQpc4+P06I2tnOK1DkjkJJeq1iC0cd4F+ypaRXgpYAYAGwga5jdFG80PzWdUGTnOKS4FG1qFj3//HJRLO5HQG6dulfwZ4UoPrdNvKvN20ZLq/tHt1f1SA3xsWevMDPqYTIlvljAa4abAOtnPXlLVRXnmlsRwmU799rgzfe4WVRzjdiR1npm41n0gdorjvo+/JxR5hCJ8xmhtFe6+ndO6ZHzj91n3SkVeX7iFduJNcXnPaH6fittdqTy/LZuv5Nuiv7wQzG56SD4eV6tPVrddX4h2p/9YQBVvOu2TuD+7iwn3pfr6ITT89it0ctzC7+YB9XV8QrE9UQdnfldaCf1mMoOC1N+2BQ/MIuwjXApdfST3QRlClYF/TJfOy5HfToROVk79vVTcLKdvQcNjhQ8rRzgdixgIuz8smqAKar8EmryVwLn5ixtCIMACHL4iECGucT+NQSW8cImjQzj6EZj0k8A6ciH9S0DwoTIN1Iq3MwUWCwBLCEFQUqMf6psL7JkJwZSwQzg5GoXge8H0waPOocGuUSqscLucgwEAEIAd7EFB2iIStMDqsb7l2DsjlAlX4MElhJ4YVJ4gdJw3ugpbyzmBmAkstCPCYigg5J28U4nKQzj/UhCBAHAAYABBMmQhWXIfLlSghrmK9nq7UMANMpo9OswfeLrDzllATQG3t+sEITEAabhCQUSdUarA/sFD5MjAq2C3t8MEzoDEM5qEQhqhSoSMZ3M9fXu7RIAbgZjDTiQgQuS/vtG9rCi/FBsEgSgBdQhEVoSefnoiURymIbQKHEO4N6ihbdLFOiSrM0wgX5QK65O/q9sguTvGD3JFIGLGoWuWMXu7gRCCaVyhfiq+ZSy8NYCcYNRGhFOyVVtFb9SenNKja2QzPVyJDVRBD8yDXGTH08mooyI1yUuD15nHFMy5eGnBdczH4kEplUq0hbG5ImQ5KxEAVcDHg7ycsSqr/4XUF1I0Dxwstj+cG4rEn6LqgbrCyG1ZrwtBNYTbROlQBpDEH+KKJ3j0iPUCwxRDJityyfSxSGU0JZnkCHxjwxRzEXWKwJw0nKLix5hMFjXkQ7EDgBFIh0c0yqOBSY7ySddIg3JUidfLOdL7ral8nupCqqskCXlMiUxMsZ+CGyGgGbAsnvIqsUoBPQY0xbRcj7ZyS+OpyrEklGATRk1jsYjJy+IRywurPhtkwI9DOE2IjoIcTNShCEo5LZnEqmc0hUTYgBazBal8zIoRMduCxz5rQFNgLQkIAAmoqM40HS8zzCYZPwY8x1B4g5ixAHYoStU8mgnDL2bEMHoEtAEZgf94uE3ctJnPTC9f7L9DsDpNY6SKyzziPBzWPC5RtDGH1IS75IKJhM6SATJKw65KTK4iLIVJaMwz2867yTP8qjAMLDrzmMJQSAQRUBNXOE/Dkc7dNBFiCoRhS8vWYsn6tJtJU8+O4MIoYkNOGCOiBFCzYbYBXU+n6ZGUaEpNkIlkEs4FLZtCc1Ap0zgvVL5E5DCX2wftxNB6EdANJVCNgz9NMcVSuISYWLUSJZsGRdEHdRx5nJANC4VLIMhHGE4ZrRdY07YatdHWSC0c64RGgBsHsEIgTRpeEzwiLdIY+h2UkD9OaK0AmIAy4kwnjZVjuA+yfEXrac//QzJiNIBb9FL/pGGFWJPSEbMexMwKbbwEMYCbGjDGNa2ZbXC4N4VT1rgenNtRGYgAgSBRPSWWjPNTIQPUCKXHUFDSgEHUmzG5Rf3TNmqDlPgzLJ0ANdEFg5xUdOlTS73UPSLHjWw0Co2ZDSCXUCUZRSVVRvW1hnG7JCtUPO1SV00XvYlVNoOMNSgDjkFJF5UBuHk+XaUYrutVe5MLltnPUIDPN6SpH0XWWEm7Zb0rupAh7uoEZELNXK1WP4E6bM3WX0SJ/do0DEhTdaDWcA0VZSVXZiUnpXq96+yKGriHdnXXSLnWeC1XcuqR/thRgrS/faUXzvNXX9WzS4OpUAgSmZBIcDVYLgm8/4SttEPLSlnsBDbgjC2V2InVEtuzWIVFLeXYizkbkA6Ah0MFWUChhekJx5HlCETDrf/ThCDoCiF4rZatFumT2YsViUvRqkmwkgAgBJbl2T4R2Z8l2auakP/7A85wAF3Q16SFk/1hWj0bCZNwiUn4PQvIho+1WiPx2awF2uyaA5eQhMDE07EllgMU0+mcDDnQpEPAjQAoWLedFfYzW6kgMcoIBD5QOun4SL0NlrLt26GbsTT4Az4QwBFgV8Pdoe9LXHm6rTToAz6wkrYqO8mVFb6tXL/tiJLggzXgDAFwQc+VlREMXVHjiJK4AyLYIAtwRNWFFS1sXcvlkQbTpXlAWv/bPZIfzF3drYsyME28BV5YacLhRSSNIAMm2CAJKNzkjRTcZV7iLYOBgNyqpd4aMcPrbd42yF5j3dnufRQ7BN/mFRyZSF3zfZRqTF+/pZ/cqF33dZSXLaX4bVYqMgAb+Er7BRREwIEoSA7JMuADRuAEVuAFZmDIkoKYCYA8oAd6AOA/gV/9BSgYKFQJAIbfrWAP4cWYxeDXLYMwGIhi9eAPZpCcEuERLku7WF8BIARQVWEj2ce6amH9TQMySIM74AOcpV+xreH3SMiVyuHrLQmqoBI52AAAaKsiGeI3cceMOOLWTQ674INCcIkEdYBRSOEoPhCkJEkqduHz4uE74Jj/zI2XIakaMOaSIr4tDE5iOajSxuUDIQkALuBeN0aQnfyIKv7ZHV6DLE6JwGWEMpNeIebj6xDJMX7d693hHq7S5WiRRoCBrpCBC13kLNnLP87dOZ5kTfmDTHiD080Dn9vkJPHjoA1dQSbklWhcz5JdANiAlkxlJRHjpwBkUo3kOwjlc90L+CyBrrCB3bjlJOlkVv5ZYK2k9qSQCQmFTZPamjrmJCnM/ZDZSG4R/wsETiiFTMDZp4zcarbhivBOZY7XXv7lrbQNTSvlAMhjckaSZNZlcmXmNnBmlmjcSCAFUsiEWd6AeZFnG7nmuNjl4+Thbb7BmNK0QcCNinuHLx5o/+uIzHOuZ0vd4TKYA4CkjRZptB1FZB6b6BoxzvHw0yTG5xvrg0xItc21gTwdaRm5T6k4aJ3S5kyRUK3S0UzgtDFSwpiWke4kIRTN6I1WroaVs0aIDl0KL6AOkfS8oAG94pR+lyxOyVDIBC7WMqeW6UExjppOtDPWytnogz/QBITjhEYoVgCQAergahARas85rpLQaI6ejVieuEsYCAGog36Q6LfmDaj+aru64jXI57sOZrTGBBYbkrAF7A+ZackAaxjqZZyOEj5AhMTbUWJssQ1R5MdWjBPNj8nuH7o2aqhxiW5+O03AhBpwoBJgE9DuEMEWD9JWnsI+bPTog5XuQ/9S5gzkle0O0VCNGafKjpzGlYTeUzFNkAEHMoAyCu4FEe2VoaUknoN1Vi2GnjhSjpmB+OvoFg0afRDb5hlmHmT5cQlByLTF1Osa+i/wVpBsKxQ2Mm75KeuzVsSsjhkHqIMRhW8EgVIxIW/Wo6zrlh9gngTlVjEDeEO2Hgca/m/rANO0sRQLKmw5BZ3G1W6E64RJkF3p8OIIRxD57ktKPGPL9hZoVkEVK+WAEQJNFvH5GFUBrx5mNvAD14s/WG8Ob4Q1toDpjfH4gNUSN0RXHmv7fo4VL4WtkB0B4ILnDPL4qFS5bJxetmv52WcFU0EsHeanrN8oj48Zp/KmsW7sBh3/TcpsJX/RDXKAAfpsMFeLIceVAfccI8fxgFTtFecENnjoGnAHA3hzOEeLKU8WOlegM75yHN9tllbyHh+QCdhMQY8PeGUXQ/9VyqLjO6e8/1NwR4NgLqBgSX+PfmXI28PiI9d0Qxa9NrxjuBkBZYBwUT+NcVUZS9ezMzZzHNckQUjJFefRrpCAR/CH75b1wahYrrH1lsL0XL/z+17xtCyzAGiCeCD2YmeLpS3vwTt1TZ+S0VzxPhCSp7Rla9cOhFWbZCdQXOf2bvepZzeF/uqKCfBRct8O1GMddKfiMmjmdSdCFX/2UvhwAXCCd6B37cD2cxePm+Z3bt5xFUQmAOgA/8cueOx4Wf4xRONQ54X3tnZ290Z4w9zgsUCf+LBA3Hu/IH2nao3fdH5292kLG68S+ZE/ivVBwNtuo4RGcU3XpA1fcTuFGxv4c5m/DuIrbUQig7pWeaxI74Z/uzdwnhFoD6G/ju5JvaLvqB1O+aQPRt7u+OgQCE/YY6kXDO+refcBZIXX+j3UaXeXqA0q1OgSe+EAXQtns6M/7bS3ClV391KYKV1Ch7CP+7XIvyeK2anObbyn5ELo9Z6Pmac0gJEL/N+AW2CbMbFGfG8za3eHO84AAAvgsWqP/LGYeyCyL9NO9Ms/BLx29z5o4iryAygP/d5gXT1iGtxGfXZfdYQbhP/scYAziOjYl/weksF3rO/bbzvMzn3STIRLlokmiBbg/w3hZaMyN/6OznMVJIVGCIINCgAheHDo/43lnX59H2RmR/1Ff/YdJYLGr4H/An3wP4oL5r8Td5vqX0NOX/FQ0IQr4PwRAAhgBgYSLGjwIMKEChcybOjwIcSIEidSrGjxIsaMGjdy7OjxI0iKuIoAcdPmJMqUKleybOnyJUyXa8iQmfPnEE6cgnLy7OnzJ9CgPgXxYRTKFNKkoTRdmWAAgAVbIadSrWr1KtasWrdy7er14iUdWmKSLWv2bMo1acis4QN0p9C4cuUW4iOIU9KknTKJcQpggid/XwcTLmz4MOL/xIoXV3QWRgcZtJInT16b5g7ct3M3c87Z54+mvEg58d3wVEIeff8Ys27t+jXs2LIJ80IChDLu3DJp2pybuTPwn3/4TColmpMmNqYBoM43+zn06NKnU2d8qUcUk7q341bL1i3n38HHDzeat5Sm5Msn5MFX/T38+PLn0z/oGDL3/JItYw4ufnx4bh2VFCmacMKGXxPU4Vx9DTr4IIQRalXbbfpZGNMaZdR0E4D/ASgXH3/gpZSBBiRYRz0GrCYhiy26+CKMBF2X3YU1ruRdW4V8qNOOm/XBxyV5LdWJiU9ZUId7MSq5JJNNzmZNGfjZOGUb/AXSY04eYpnTcJGQQmIn/30BABUh+zh5JpppqqlVL0gUsQaVNc604ZY8aYklUYgMaMpepDRhQAAGWPAIPyuueSiiiSrq0IzaxZnfWjnW2dOdH9YVyIimkBaKEA4MZMEogi06KqmlqgmllI9uZ+WkP1U6Xh99ZJIUcpjUgMFAJdjCj6m9+vori22+qapuc8rBYauuTjqcJMahp0kjBkTw1AgDGQosttlqC10n2DlK7H7f6ZhsUK9uFkhRRxXIySAlBBpADcpcuy299dprmDVppAruWfwhS265PRIlSCemLMVJXwM50MQ4/dz7MMQRY9UmEHDya5ax/wIslLlCfRbaUpcY4elfZ7zTj6gSq7wyy/8VdRvFxWZFyse4G/sG4HCTGKzJIDEI8BQFjxjgT8otG3000gTlu2/MLflrM3Ad99RlKHuJ4S6gNbjyz7xJe/01xBRb3DRLGUbWB9TjSa2TW51w0ogREwQAgABNZNM12HnrnW0oQMBMNkuW2ZV2h5zxEYiBg8hAMnvz8Lo35JH7as0aOoQBuEqWzXEl4R92HOskBlwdKAAjjCI56qmPSjHmKJndBtqdCxxXeYwQ4cDc8BaDt+q9+x5jKD080XqVZKQxuOx4lmsXG1iXDA/vv0s/vYOU93A54GuVsXnyk3pYlwFBSPAUvLtSfz76DxLzRMVkewd798n+x4cBIpA+ARf/Dae/P//wBT980wS3tviFhydsMICnACUDSPCDaP17IARlYw055AB7/NIe9wgIMLhIAVBPwR81IijCEbaGGFFoH7jeFzsNbowNMSBIAGTgCoeRsIY2LMz/Lmi8O9SMhfIjgtwGMoEzwKOBNzwiErNiDTdYjlgz2R7nfJisP3RQAAAYSA2Q8bgkcrGLIDEhCqmEoxVKEUuFGA79DPCzp5TAAO0wohfjKEeM5DBO/BlgGV2FxhAZIAbj++AZ0LHFORKykA+xxh2uJ0YybE9jeeRMIH4Uq0JIYhK228AVmdOEYqDMkJ78pELAmAYbmW0NZHwkXf7woxAdYhKcCAUnJiEF/+c5QGsOKxooc/nJUgjPRndEpVwEMZwf/YERmejEUWB5iSuU4Gd068Aj5OFAXVLzk4hUpIXm1BtgAiWSfIiVICKhiU4YxxSlQM4kxDACAeSuA4RIB9eqKU9PilI/OAIPN3NyxlWizZV7QspSMtEIZrJzTCPIgzoGOc+FyrEVQAAgd36ZT2HGKkTGfKVozEkaaBnBfrkbAR7YwQ+FMrSkSURkDr7AHW06UorC/KZdxEnOjAI0PZ2Am1OeIoAYINQAcDQpUJN4jBOOMjf3fOQ++fDNVmKUpgBFTmgU90cA1NISjsNlULNqQ4dClDISZSFFicmISzTVqUjpRHpCIQoDrP9zIH9pwgzjqdW5HvEOKe2OhrZJwJcqNZzjLKdZn5oeVDhCCPYbSAAowAVl/JSujiUhMqwAhKKi5Z497Jw3lzoJTfwzsGfVRCbUKoYOOOCKapRBHcZRqMeytoasAAISKrPDKKbtjEq1qD89m9GlGIgUQiyIBISgino0trXGfeAf7noWlhKOrzE9JmB1m5RzpjURcQuUwmSQh24ct7sjTIYWJtsvcdkss2iTBGel61RYJo4Ih9XpCM6wO+/SN4IORcLYYMKqZNlWqcXMrXppSt1LvMEIFpgWfLngCgbVt8H9S65KMUSTY7XKuYIw5kwDvF5OJOIKMrDAGgGQWCeM4h3/A8Gqg1P8O/CKV7/kxZI3J4nezmpYNOhhAxFKIAHsitgCTRgFPFUs5P29Arb5XcnTLLXH/5a1xk5NhBRgIIEQ020EXFBFPKI35C2rThA6iHDZJnzK8AzTotB1cmCJVL8Ekm8DNqCENPwhVy7TWXrg7QFlM/diSEqyD36lMZrz0ok3EAGBpv1gDc6gixShuM6OTl2RY4vkHbb0J0llJYADvds+GAAGB6FqB0g8Djlr+dGmhlwgvqyS1415KKoU6zEBreFSVI0TksAxDELAZoJ46k/ZUFGpTy1svd05zwIMSoxj+lcnh6LW6TGAJhIhBiLAYAO7fooDKGCDMxjg10Nr//Swww25WxRB0sVLQwZ5cul+NtmstHZbeuKtCQN0IhFs8HAJNrBGtzKnAy2ogyuoIZhgi7vgehNEDsbCaju92sztVgp75Z1WU6CCFIkocAzyve+njKnfTQC4aodWXIOTHHXM0AKej21hmZbz3VCVNydKgYpWnMIRieCCGGowkB2PiSAdd4AFZNCEM1iiGPP4B9HAXfKlR47cRWDkHGC61EpmAjmZiHfVo/0GNojBCESIwQhEMIEpY/fTApjACGzghDqMAhjrGBrcmS539IVABCzodKdhwIISlEAE+d4A4CcwdgewkyFjEoAELFAtA5wBErqghntIbag5z73yvzu0QV3KTr6Oc94BE7BAB2TQAiecoQ6WsAUw0DGPgdiDHvQwQOtfb/nZ908XBrBBE5rgBNKXvg6mf8QoVOEKXRRDGdJARzochnQVYVXptH/+/rg2Up+OlKSCmSb0sz/HgAAAIfkEBWQABgAsQgAaAKEC+wEACP8ADQgcSLCgwYMIEypcyLChw4cQI0qcSLGixYsYM2rcyLGjx48gQ4ocSdIgvZIoU6pcybKly5cwY8qcSbPmQW5BbOrcybOnz59Agwod2pIYEIGE/hFdyrSp06dQo0olWooGBoH2pmrdyrWr169ghXKbo+KFAQFh06pdy7at27bHkNBAIvARv7d48+rdy7evyKo4tDgw0EGp38OIEytevHasCiRaWABgTLmy5cuYacZ9oUXLkwgGHvnLTLq06dOoF6KigaNLFysCZaSeTbu2bb7c7jx2bWUHgMF3bwsfTrz40s1WeGsJYMCS8efQo0tf2QpHa9dRwsQOPr279+/gGeb/3s07DIzf6sKrX88+OnLXr790EWhpdPv7+POTfmUdfhctXZQhUA36FWjggbjphoR/naUhkAPqGIbghBRWOBUzUbyQHINpaCEAAPVZKOKIJP7E33X+WZEGGQLZYF+JMMYoY0rjLejfa2kIaIAE8Eg4449ABlkRhhre+BoZHRoAgCo+Cunkk1ASdKKRR67BogFCNBnlllzG+IcKRVBpRRhtkMFCABLMo2WXbLZpIJEb3qjFF2VKMdCLbuapZ3v80UDlf12skUYagzWx56GIghcImH92tsYaZbBgwAb3JGrppcTB+ed/a9TJXDGYhioqarLg4OemKpbZhpJcrDnqq7D6/4XIC2GimkYbaaxRgkD7xOrrr3sxY4UKUcj3pxVktCGoncA26yxbpZ6KKpllzvHstdh2dYgKQBh7LLVkzCGCASVwl+256AIlLLHeUglgG7iuQQRz0ria7r34unSLDtJuCmCnglqb78AEwzQrEJvC5yi8ZPCxgQEjFCzxxCJFowW7CbumRadl3pETxSCHrFEuPfSbcKq4urFqaPaK3Jc4MIvjSyelWONybQdnDJ8VZcAraB8SBFBDyzfvFXMnjIADi8BFmxZNFxjrjCy8Zf4Rw44RNr2YOAYcQwM4BuTyitalkWzyydR2DIaSTJKtGMx8pBHNMWC7jdkkLyCsM290wv+bhhx9CNSEuXa/LM4TX7wSDBnJFE5ZNF+8UOzegFItqCACTaCm44dxXYQWXLMSBeeKmU25xhtTTcYfUoBWDOGk4wWzAVH0ATMvEdQdO1946035wgzzcYdAXOxuU8zPhALR7L00ftDsufgi0Ox38AFzLlcZrxfkUVOOMq5y/CGQBb1q/9A3zHxjkDjdOGMRzNuE8cxAXBtwTUKzF1L37M2obwDM4FgDMf7HtV88gWuhuJL53GK60/HmVj5LgyCIIBBpLLAh1jjEHzQxpmZMjxll4MMaXjAHbEhkdnLoxfRgdgwk4c8A2JgEAcERCkFw4g9rYAYBr2GFbhBQHKGIRCn/1hCNC7blEnlzIG+S5TMyBEJgeTDiQsTxhVzA7BsvwAE4xPENMkQDZsjgwBJ015CY+UILP3xFGiKBAh2tj2ulSMbsMmFFFhZhizDTBCNWaABwbEOKa+FeFJR4JKqViQ/iIwwgFfKNCHBgdl+YjDhY8QVnxOwOAKDF8rgWBVwQZHYvcGNB8kdAA/QAbLPTQjAIiA0V3G+RbWmgEsdkyHAVIieDwRMsCcKIS8zuCQH44iEAkEgDkAIAe3wIzJjBATKCUpSfNEAyTPFDDqCilE9ARim7UIpdrgWJvnOgFfrmtzUcYm3N8YcuvUk/cUQDA2uAWTLucIz6XQIAynNIzEIx/7pPci2Uz+OaJO4XMy1EoA/cEAcutPNDSTCUnV6B3AueQMjKWY4MgggcAITQj3VC9H/HiEIauhGzktIuAu7TJ8zWUEw+AvSN4EDECsXBjBcA4AWI+IMPf3gLs3yUK8EAwtl+lzrV9UEQMNjRPGD3UXBIgg9R4AQe6wezXwSgE5s0wBIkMcp/QrOUuchFO2mqAgAEoA/+I2AwcvdTrYCzohrjGMPmcIgrTEYXRPMmzKxBrG/UTyDXwEEh3igOPnQhrXzsQT7baYCX+tMAh9gfzILRBWb0FABA2OkyAWCztkKFexSF6/fidYjhGaBVnn0jZAGQib+CowuS+OtYUYDSrv8aIAfUfKxj6QdDGa4wGjT4ojhyg8wVOgMAKU0tU4L6Arg+0JCQEkQiLaDcgcyznlxDBQBAN5A5oAKAHhwrM1Zp2yJowra7LWUc+agJgcVsDktY4TEA8MfqEuWtzp1aLftwCGZZ0L6YjCcCAdCF+kXiFTEjxmIZMrsuJJOx6Z3dIRgbiETGLBfxJSAvOGDfoUSDDCoIrWiZqLo7HEIOAolihzkBBGJs0QBzAEArYBaKO9zixqgogwpVagBBrAG9CpyeAZjRzZneAgjqm90kOPFDVBShw0FhrnPLY0jS9mEDk4FyH+fACr/yAgN3gNktIgCAMpu5iCoVxy20+MNpHiL/AijgBCsIKtBXzrQPZHiGOGjYBlRyrQ2Y03JP8OvcOVX5coIIQpa1DDNa8EEOa8AF18SBjBtb+hae3GQ3SDC/mDHDk73gxY3/CMAHjzUYfJgDH27BWByQV9A2iUYaJjplQMmVYfxd26JhDZL8mRoh0BOrMm9HF17XhLkAmjLwSnwIPkjA2CSBGTeKsFP8cS2yWe3CAKEtE00ksdacqvKtNHgeboskZrdgGrB7u8lJDtbcL4nGCEU85dEqS4L9hfdIZkeKawI7u84rIwvnQEZ9pwTZ4O6CSKtctUOo2+AfmZ0O/22A+Q17yAWHOEmqEs56k5jZfxiXAfCh8ZJjRt7k/wE3LcXthkMkOgBoMbnMKUOMInAm4a9Jm+XSUIhzDqQfMw86YjiOc4tWeXXNVpKLhM70vXBjDSkH978Y3rBDBCKpBohHXpvO9a1sJtkJXzbV/naI/gYgAHjtutrVUpUeFF1jEIQuzx0ukDNsfe14X8rTo55weysL6eIDwAjKl/fCS+Xrb39Nz6jOh0Pc8izZuLvhJ7+Ttie+kAwvA119nk7Ke54ojrHR21cu7jSUnQ8CEcLnVx8UxCd+nAz/eyBcnrmlsv72lWfN5Y1+9MabPQCvw73wZ1Kj3Yudapov+xwGc4bhOx8m77n88ctZ9kR2AOjPz75KVoOiy/tdWWUQhP/jrwYhpmr//Bwp/u5xFPsy8dfnABgF+ucPkujvXr9U3zzqDdAEydP//wxRHd3nfR9nSLlSdoIwLhagDwDYgBahfutHeoeGb4fABAAgANJgfg64gQmBDEhQJOsXH+3nfmW3MknBgSjYEFMSgrx3dCamQQPiUSk4gwYAges3fdRXdmfybBpIgxuoKSwYbrF3OWUnBcyRdj6YhCsYhN/HMH9QdncQc0mYhDYYgk3YMQj4MBEzhTTIDE8AgkxYgAbYcmV3NQLRg1yIfksYhiNIhOeUZTKYhvP3JbUShDk3gg13CIHDf/4nh7cHhHZoaCNIBi94CA9DXX7ogH1ihwoTKCP/SHZllxNSmIj/tyh1GIhFNYRzdwhrwByUCICAyIiZGHtIZ3VKkiWfOH/RwogpEnek6HuHIHKbk4raNyuXyIgL94hyUHaHQAQXqAy0qH3rMjmsiHntJyg9dwhsgBR9GIxqdws5MFRMqHMMZyWz53gt4ozPlzPFSGV4WHVllznxEIfaWHjD2C6i2AV4WC28aCcBAAzkWI54ty/SGIRT94jmVHbLaAB50IzyOHMHg46iOIrV6ES8OCD/uHoW0z3dqAWu+Iq8uCsSEA8J6Xmy1I3P9Y2bl2/viH0VWXjciJG8sXj4yIumlQfx+JEz9zQM2Y34d4zhd5AGIANoqJIyd5Ei/ymB7UcG73cIZzIBFGmTa9c7Ipki5DSIhWiEHSmUXSdIAlmM77KOBwiFKVaTTKlvOCmSOCh34heOM2mVV2luRFmUCkOQpNiTu5J1YBmWvOaUZLkzD0mKSXl2wMCWQZeVRXmFY5ePpZVi/miX1UVob6lwJNmGFFgIzyYDfwmYniVIg7kzYkiKT1h2EsmYJZcLQvWYO0ONO5mUBuCJlmlwgjmYsLeOKcOLbHB2oWlwEkVvgxmV64iMCDgYBpCSq6lcQVWPWmmWksmLIhAAD2Obt9lWo/mYvEl1hUhBBgCMw9mWIOaaj6mXY7eLZScGniiczclOUqaZrWia95aMgSA42P+ZnbskCd/GnUvknXloiAaAiOTZYc3wBSiwBJ1Rn/Z5n/iZn/q5n/tZLIPynwAaoGEwB4FQoOQnDZXyntWFcOjpjaaZBhupa6qgoPZVnJopiA9qemW3h8VDoaklb7TWoBrjiKZpjTLZAePpodrDoCK6lYwXkVizmCpaON7mdiKqMLemke34maAyoxAlayF6o+znnVOpjCnmoxDFojeaiyUakzBoAEODpN5EdEKanurJk7z4MBtwElK6SCgnekKqkzpahhf4X10qRTV3c1V6h+pZpFfANh55pgtEpWsKmyVKgabFBXEqp8azd2AqpPfonVjqcs82AnvKp7HjenV6nHL/CaMSkDWIajyWt6YKE5clWXYUBJqRGjuhR6lwqZ731pVggBYqtqmkExc0AHaUyjOgmod8MBhGsJamSjGA4amQ2aqEmIUCwaWzajedaqts2qZtwItXgxYp2qv5gqqquqpHeaddaSeTcazIei/cB6wj2qp56AaTOK1NU4WU6qKNqofPFgOyyq3pYn+2Cq4MB4kJ6Inlaq7ZUq3WmpHqKZuPJxDSCq/O4q2eKp1UN5noZACHqq8Eg67AGgWFKai+JzD9R7Ags4bW+pJESp3OBjHv6rC/wq/9ypmxOXd/ICmPKqMYKyoeCIbzWpqgaqKJxqMXO7KvArHWaqdX6nu6Rgj5/+qyoaKxnqqumbd5bFAoOJsvoTivQgiqg3J6AjECDBi06QKzJ2upeGiigReyTHsuOmur/vqiKwt8VXsuQ0u0EiuodFUIbxoaN9u1euK0JxuZUpmPP8t/Z4u2bmKJRGskYtqk4ncHhTqwcvsqX0u0GIqr/LWHPCKyfbsli1i3ckKiuGpif6BoAsCch+srtqi4csKoJSkIUoAWnjC5vkIkxGi5cYWtocqJxOO5sbKKotudpIulbPBsAou6r1K5qwsfTNq4zTYuQCm7onKOtWul2HqAVqOpvHsp0KibYMuxzqq5xTsqhUArv7uZpOuEhYBOdte8ieK70SuzoJqry0e82P+rJ/QYvY2Yo23acg5jACJwD3EbviMSktuLuYaZUUkFHO6bJwsZutsLtaa5On+QqQbQsvc7InhZu1krl4IABpPxCIY7wBMCv+TLqtNLdmsQc3zrwEHCkvobvWFbrzxnWjJAeBgMJQVswMorqH/AB7oblCMMJWNJvjvTrLiKSPUbwC38JG4Jw9c6vbgyBxP0ma5wwTdcIiVcuzzbfoOSwH45xELywjr8H/LbhquzMobCxEDimE/8qTzcMKa1hVY8I0X8uwdMinNwZQYAlAL8xethoTCMsDwcL4HnAGaqxiSCxVkMvKRrJf/7mbZAxzCSm3fsH9nxxofELDbrxyTCxjD/nByEHC4/xoeILCKgFciNyL8Pugb7KBvtG8nPAciUjDqPwsOCwgcTMCnzwMkVosg6nDrmO7O7QpuojCAfFmKfrMWt3L9zcDWgkcaxXBzbWctDqiykmytGiK+9bCCqrMMvecuGGbCbfMy1McvQGcgSy8w7+cg18MzQPBueDMzBajm42gbPVgJLu83tkcyrLMPC3KZr8DDmjB8gOs2BHKjQVa+RAnPv3B5KWstbac219DH5zB7e1nHAzKj+zDBMgBTaHNCXEc/ezLrH+KDM0qEM7R37DMy3O4Sx+QUDIsQVPRx0+tCEWaLryFAjcMofLR1+KtLSS9KDKBCPmtLScQw2/7es3ny3Go3E4yLHMh0dk8rSLRi1xwgDERAAQdzTxsENacB3D03PsTmESBAAAEAISG0cisrSRwzODMcEzIFaVQ3SNGCjQA132Gq+8wGlC/3VevGrYx3MKVtlOjIC+aDWt6GsbT2S03trj7wBLEzXs1Grd22MZW1IMB15fj0bbN3WOO2dcjUukZvWh70WVz3WKJvX8AIDzGELkB3ZaSGvgc29gw3Qm83ZX3G13pzVbTjRpG0aBjvWqD2CDzXaq80Vnh3Ybs3DujZ4s40ZiR3YY9yGz+bOu20Zrd3WbkzI8PIgBtDAw+0VakvZbIutji25za0Ypi3Si92qWFfdlFGycf9i2yKI3G0QA2VmACLM3X7x3K6tjuL9MaWK3n5x3SL92u2X0KcF34jxt5990KbJLEYg2/gtFOqt2JasnhxNGB4d4GrxJX8K3rc9vVdSArOo4Hih374d3d2bOZBK4W+RuA6eIhh+pQ/D0xyOF3T44TeS3VcqEALQoyXeFhb+2eqMrWSQVAEwoS/eFh6O4iPK31IcBFL9CDneFrTL46BMyFbC1fzIy0OuLhmywSgexVFLBsXMBUze5D6hukauxaJMBuikelgeFkW+5QpX4LFJBo8cwmH+FdpL5sfNw4OyfBCTDwC+5ixxvGSeIidMpGkwPADgnna+FRC85SoulX1eyqX/XOeBjhJtTuagbbRr0Acit+hbMb55Xr5vnCt9sCvGSulSEZCXfuRw3gZ/sN1X7ukukb9PaeQO+cZkED5m2Dao7hRhjOK/fXQECuSTwdyzvhKMcJ6hrniu7sPKWZu9zhQavOpG3sFXegfMawBnoOjHzhG1but7vo6EqLlowQXSPu0a4cTBXtmtaiV3UL2px+vePhJf0JKX/ugl2jDV6wAAIAPdnu4WQQs4wBn8ue/83u/+/u8AH/BWoCIBWvAGPyhhcAc4lLR9be88gc48Tt9N9IQCc9IO7xOCZNOhfusTT3ftqXUXzxMNpPF5zvG4kgazZ1obsOEhXxOERvJb/uat/5rEfbkjNtzyNTHJjRjsgm20+LZ/EpANCY7zB5eZixvu147EayB+++cAylDvRL8QFgrzHy7u7NxzfMDpxQD1UY8QOm8kVA/e7o6H49ZsSSUAutD1MtHN7tLuUn50FKskRq32MZHMYb/frat/inbj6E73ECHNOnP3BJ73pycbnef3LPHLGSP4QM3xudpsvggAnYv4LAHxI2rkMt/sGzovlN8SgH86jP/QzC7FL9gHnG/snY8Siv87PF7ohwaOfWAEP5f6KDHQFRX6/DzjUvx+rOOJQ0/7F+HQt//hTn3mk/kHRtDVXO/3Fw36Dv7akHL8V8AcTbD8dB8KNEDQSoT78//s436D8tVXtlUM/CGBcksgdeD92zSvQWUbu+QPEs1PSNz/xBlt6F0ZCFeAFkJg/V0f0gDRReBAggUNCtRyUOFChg0dPoR4MEqZNhUtXsTYJs0aQYcOBbpiwIARkSVNnkSZUuVKli1dvoQZU+ZMmjVt3sSZU+dOnj19/gQalCe3NSqQRHyYEOlSpk2bWgmTUarFjYU8CgIjskY/oV29fgUbVuxYsmXNnkXL81iRF0qdHnT7Vu5cp1CnTk3TxuMhrFq5pgUcWPBgwoUNH0assxSNHnThOoYcmWHCu1LJyNnb14CMv4k9fwYdWvRo0oOJGpVMMG5q1nQTrqmMkcyczFn/N+MrnVv3bt69fX8+hqRta4TEjb/VogV2bIuza4uUgfv3dOrVrV/HnnJx4+Orj393qCUv84pk7uwtZDv6v+zt3b+HH58stzmov3sHn9+glfHkzaMHQ4DN6mFPPgMPRDBBBUsKjgb8WHtQP/2soIi8Nf7bK8ABF+SwQw8/LG0xHCSMUMLvrCCDvDYu5GOvQzSUQR4QZ6SxRhvDos8+/Uo0kTi7LCSjDxdhJPBGI49EMsmWghvORB57TM2KL1S88I8hBZShHiW35LLLGVGhYUQon4QSMsosLMPKDLH0sk0333Qvx6PKJLPMuZJbLrY00gjExSsiuA1OQQcllDQmrbCz/4s6E20KT/L27GgvKUrSp1BLL8UUMDDFTHRRRpfi79E0IvVoUpH2yTRVVVf9iZs7dOz0U9ZCZW4jUg8x1QjpWOW1V1+XFA5RWT2V1aGJHl3DKklL6uxXZ5/l9RUcOB22WMhQfFQvF6UIwAAh/IE2XHEvdRXWaq2d68fYLnPxECJEEqLAceelt0tkonhBWHSJRbcgKf2jzcV3DeCiXoMPPlJaavft16kz1w14LwMAIHgfcBHGOGMFy52zYUU9XsrRdc9zcWKC/blYY5VXbo+ZJ/IFubiYIVKOOQz3CmJiPFjmuefqFJ5Z5qAZovWuC0neK4YAAMiDH5+fhrq0P8wFmf9fdIueisV2J2Y65ai/BpswZvDVN2iriz22sguFdBEGAwJ4JGy5505LWhqGVg3vg7BVmww1JX7bEq/pJrzwoAJRoQi9BzqbUXWzTuNvj0oQSRXDL8ecp7FhXlzoxbWYsrI9+3RRBJFcaTZz1Vd3SRYc7u6c8dhfE31UFwvZAAABdBmcdd9/FwmRFxSPXfbPa74LUhcFmUAkYICHHnhmrFAhii+KN15v5PHiyMU/JDDAAWWiJ3911xfGvvEesc6oKhf7EAn88ue/XJDhsTdIfQmj6E8qitplgwMM0Lze0c+AT5te9a6Hv7wNLW1TYZeL2CAgC8DjgBf82i1yADsGFkT/f+B5XEac4yIxdMsA8ZAXBlW4MuEBoYOPmVkIZRMxj1yhWzK4RwpXuMODJdB6L8zfzB5mGaSVyi88RCLCbqEDDgLRgzGjHQSL6C6RNCGJV6xXC52okA+2RmQZOVq7lAYALqQOi2fsVTS0oMAtwrBh4skaGVrUtonlAY13/FUuetDENj6xYeyzyNq29jZP4NGQrJLEC4CwwD4GsV8UiqPkBCESAVjukJe8VDS+8IIfNtKNxeIbGCP3PpE4gHeYROWg9MhHT/oRlFGRyuhcdIdSKqOAqcRlkiahyFZOxlr/iqXt9uIGkUwgGzrMZTKNpEk29pKLxRriRZS3JgNYQB3K/8QmklbpzIZ08U7bk2b3lmWAEcwjm+e00SV4yU1fMuqL4VQWX5igFXTWE0TMjAI7u/kpQLYhRS4KRBBMaE+CcigXQHiBPsPjuP41BzN7+QMMKHYGZBbUou7ZpQsVuk87PfAiI/QIHyhnAE847aIndY8mObnRhZYplB+l4RxMhlKaticYQGAlSz8pIRn6E2mFaMPExGfGmhZ1N+rUqE45aiJgygZpfQnABo5pVKryhplPUGpSehTNFd3sDyExwEirOtbS3DSnWd3pfcDZ1Tkeog8Dk0E+bklWuhYGqWilmZPyVBFBhlSgBmgCUes6WMFEgwwqwCpetTqhhlZpL7TUGf9hJWuYmyZUsXnND/vWMEq+0DIAAfBERSc7WrLc9bKY/Y5HNcInjwSCDc1zQDFMSlrakiUaaXhBYk+72OO8dLWR+oMY4icN0dbWuECprDcVqlyFhHCafbBNCdRR3ONWdyeaeAF3dhsR5uYvdFQRJx+4tRl7WNe8QInGGnK73ZAZJ4oVcV8h+CDQADSBH3M9b35lklz2tteL4PwnX/gAg27lgbr6RbBLQkGDpPaXu8QBcMD+cIcNGKCS+E1whlOSXhUswcFM6a5AsDbCPrBhYhIoBoY1vGKREIMtIVZqiPn3UZLxwTYWMICKWazhBTf4w0jprkcxxAcj3HDHR1bJNdL/QLUfo9Yxj/uPIPgQA4oh2conYRKM8apcdYUxEHcYqYGvfOXtNFku3nyY1v7ABvnZ4sBjPu9pOmbmRjkmioKEbjHJMVs4rzjLdD6za5DnWD68zQAxiAef+5zhMgM60Okaz2b/UIg7KM1bgl10nOszZ0fXWS60Gp0gwEwxR+g407VtkJb7qz5I/ja4pdQFpk9dXRF12s5v4Zut+nAFAUlV1rOuLTfkwGRbg7gusLRVoUWC6DcDe7R/LrZrnhK6ZI9gYk2wmLP1u6lom8nY5ZGDIExMyF9re7Ic67a3gVyz2fwBDAJEsanNXddDpVvdNGP3HP5ABAyEdbrzNm8rpmXv/8h4KlTmmTLFapBjgFcX3QS/t0MOfgcKi4SiDa9uvSEecYYcyzxpKIkr5I1xqkorBxuXDJlQdLSBiWTkJK/pw1HO8b2F4Whug3l1N1e2mUt7Ml+4EDFzXl2g9TzlvkwDGWzjAFsMnbQyNzrNGSceMuQsrFN1umR3HvXUREgLZShDSYSgj2ZnHaVF5/rR90YGU0WAEGU3+0mnRry0q91fYQhCvyWgjHLH/aRbr3vXDRKFMLCgWzH4t9/parfAe9FfWuBaGRVfV8TRvfF274IVkCDASvZ98gQF/OUFL2LDVzMbL/98Nl13VtH73Apir1Tqqyo8y7f+6E8QkAHiJnuq+v+QkbZPOc4nMD7eG1XgrAc+XUpSA3MWv6h8UAEQkjN96lff+tfHfva1v33ud9/73wd/+LeP+5LMo7zOR6kPkw+h0htAtuin6RLvpurLWyEKIRgo/GmqxY+t/06Imqe3yQPU079LUqNmoj+j0wIr0IIwIAMYAJThI8ACNKRtaiD/QworkJKNCIPlo8CT4r8LxMBuQhQyWIMLCUABKKQPLKho6IJmEsERzB8GDIPxSDoINL0JZMEzwgUmWogELDYN/IL+SLoyqDAAaIJ82EGCyqh2kkGEKME8STo5MBUHUIV+0MElRCKV6qRnksEFbMD+uZA1ADORGIF2yEIt5CELXKr/9RPChmqDMkgDGxOJADiDfYA7NUSjJuQt21vALjDB9jGPP7C6eNPDbMKn33PCywPDGoyl2QiEOTAdA6gBejjEbGJDJ6s7DewCOOSrFAmEP5ACBwgAAVG0S0wlPgSywAPDQJSKC0mDPpAvt+kWrENFXMInT1NAGvREqkiD83C3meqHPLzFNdwj5DA6TuxFqpiNQpCvQixGZTIth0G5VtyrV7yMjlgzk5iHU4xGPFIp3aJGe1tAqFjGQDJBK5GygbHC+/pGVDKrW+s2ZVQRMphDj9hG6LCgd0SlabyTYrNGFUk6kpGyK+A8QkhDfiSfaAiD6ik4RytHRxTI2YiUPoAs/3IyAM9TSBX6BYQavSb7w3MEo3S8Cj4Aqwiwo4TcSODxR6k7rYBUkRWRQ7ZxqykTiRIgrpU0JMNCLMfrLw2UyJjUiJuRMuF6Gy7wB2LUyfmpLPfarj8sg2t8lGbciz6AH4oRAWnQyKU0oJb8SLSCSaHsKjcgHQETAwEKAC7QB5XkStW5rfXqDrwCSpE0GnuUHD7ogxgQiQ2wpbZEo6YEDyDsF6iUypjck7b6CD44S4qRPL88I+zyMadcLgZ0RbFkxjuIp0PASyozgApzzDNKL7jMD8F0nA20TIy4kHADKDo0tHxQys/8Hf4iEWfixMo8za5aA8nRzELrlhLoS9hMov/FiMzA9KRGLEzLlEPEPIQ/ILJ+E4AzwIfXBM7V4TBO25E+esPbFMQp0kw2KAGK6QBbnM4VcrEmGRMn+kPb1M6hnINbWU5lszBCMADpHM/MEU536iDjXE/UNMGy3AvFbB4AqAF1YMv6nJttWDLr3Cr8yc79vAhYpEmIoiWKkQBVKFADlRtow8/OSc/j1M6BbJeQCgkAAAAhoIcLxVCwabRzGZpGpEvLTLo5yEx8ZINJFIFiSNEVkjOGCZoGdVD+JMsQpbScKUU88MYcLR8NhSYoisIfHUnOahd3k4AbYgckVaEV5dF94UUnDSbl9AhKK4HPqlArxSBucANi+xTSfAj/H+VSaTKPGcXHlmuCeyDTC1LSwRyWJm1TjIhR9/xSMPBMykHROvWZWhOiTtnSPeXP3AzRkMI5B7AjQjUgqKsalzJNRc2ImWxUeRIgADACeDhSSfUdjTMbJ9FTTHVT7tyLNSgJC3g/UZ0fbtOe0UxUVHXT9tzUPygJB5BPWJ0fSi3V3kKUF3XSC2kD/2wXqwsAIUgHX50fUp1VL6JMD1VUCN3UFymJEbClQXVWjRE49MGb7nJRW8WLX7zWUTMACdi9bo0eYI3WJxtWcsULqmxUQYgBAREALmg+doUeZAgWBvqgsJTXB83GazUCAbghdKBPfoUatMOfxolIYt3TMdRN/xKSHxzjVoZFGHednUaJ14GFoHvcVHR1gEfAQo2Fnn8FIk8RWJBN1WsVBJzLP5T9HYd9ITKJWJedV1zdVKvbjNOj2d/h2IclwU7U2Thi1E2VAvkpAV1Y2KDlmdDbogdp2aPlK03dVNsYIEuYT6j1HZud2v0wR6udV1X1iHELnzqIPa9dHXDog8TpJbcgTLJ9RJ4N0RrtFgFwgnhgW9+R2kZKjmmlW2wM0kaVqToUAnUI1b6lG8ajTcKT2IGFxYo9BErTy6WpAWlAGcZdncrrpdqM3IE9zHqdr2zlu6fl3IzZnC6cWsFdkcHdTjhdRwEKK11guNTNHFh4HU9i09eF3f9PVM12CQQ+IAL52YArxN3VOQS4baMOfcXfpdgQZc6ljR8DWNzkDRvfQ08apFZqldfklF6TnFLs/R0NQr7O6d2soVsQbZc+UMxdPQOyI9/MCUGAPVXm8F5F7dN24YP3LaUzcM35xRztfVjutcz85dIx9E/56gOjDJ8zkBEBxhzz7aD0jUkEdlBrzQw++Cr4lYetlGCeqV8Ovd/TxOD1ZN+r6AOQGF8HgOCTDeHCOUDWDdda3c8TPs0YjafhFYQrkB8JyIN72NwYLpxM1BsLXk8cFsqgK0vmFAQjkAAKzQMlJOLLGWGzKeFirVa7hCg+EISW2wBLQJUqNpwZVsSqseH/at3T0fUI9/0DGcg9C7jCjCVjZzHimEHiieXSgVSW/u0DaysJN6vjy1FFJgVEJU7iH91fBkZbAJABHEXdQaaXRGTSMCRbRF5UdeTgK/BMARCCnJTkIj5GPL5UusVki9BUKXsr+XEALpiuSA7leUmk4czTQ/5d34XR/xhetB2gPJARWI5lcaFkLbXkW+arXG5P5ryCSQSAERgFEA5mlbljfvpYYw6kJU6RP5iwd6GYALABrYxmw/FKRNUC9bRmXI4NCO2DNgDkdD0DdxjicJ6bXMzTYj5nMBKVOxAvEci9EvCEbJNnwjmo852Qar7n5xWd2ZiDGIjikwDmgBaXcTbV/3I+ZVOuyzXICkBJVy6oUoiW4U0Sx62ay4OmEmwMO0NrZlVYS48unI4kaPcyaJLGXxGap9xzgCYAWpYuHIm+D9eV6Qv+KJMAgBIghF/WacLhyZAOzJH+6QOuCCZoHkoSgm2l46PuFcAkkZhuaqEEA5ybmBGgBDq1anFeJ1ql6K02YYYuCQng6KQcaxluSKX2kbFF6yVOAyI1IRmwhap+61/B6u5AlKisawtJOrZDCUeA54fu64guawjz6cF+xcK2x5Ne6yYQB8VebGHGLbmODKaGbD4tAzIAOzK8gzmAao325MxWHdmclcCuaFNOg9C2xza4gz8IReL1zLeRAU/YFf/VNhzs0q6Ue2y03izRtsc54OCOEAT3JYJJnJgSyINX9u3LCU3O/rRS3uo9Ee2onIM+kBzmvAMpoByKaeYzGId4nu7CYW07y+Jz3qzCzovadk8p618jMJ0qK4EzEAnMTu9xuU8zMWCS3iywO+7kFlLmxEsi2AABIG/KQYcc4+v+/pXqvBbsNubiDu016G7Kba3+1eZ3+ay3aWe3lnDMKc+AbW+rfW8yEG3aXuFr5Qv37QNGYAMipSQZeAQU4u8Sn5f/Ro4AH9wBD+00QO4/8FPpdd9AGIQroBwTcoAaUAVL5HHW2VFcs3CdLW4W13AOhvHlQXBGeAMFzz0AkIAmsAX/rojwKYeWE3eYFLdV7W7xO3jxLvee/hWEJW9nMzQAaVBz6MFS7kpjW81ye5QD+abzdlluvEQE4Rrfk7DE6+1zupETpsjjBJbsLedwOkdwKZOBDdBoCxMJctjxSEeYVAMyN3fQ2GbxjZBzZD30zLDKbVxwEwqAEigYgCZ1Pw8TQLdnS2dxQjf0V8/V/r2CGODnkgCACRACVeDbXC8fM0VTf7nyJL70IodTYYd1NiCCGIhqkUh2IbAEdRjGUXd2jDH1ojXnHA7t0dZnV8f224GunKFdb98AcL8mf4DhcicfWf3BQLfrX0+DQjfyd+9y+BkglBipfdT3A8oRD1uISleb/yIs8EkjeBhPjxjYAAwYKDKvgTy4BtzI94WnnztlHFSPpXXn7jmv+FwtducuCQEYgSZQhXb4h39Ab5E3IDDRAbjw98iW7fge+JXf1DZ4F/DJP5hvAkKQBqe5eZxn+E0TW6NN59ge8iI/cmwvBEHQZhubpwqbWQGgnEcYH3d0+iSC1qr1RePWcO+ueEEIRff14zmQgiBggTo8iQCwgIUbBcWdT3Iv+7AxuZLvdb6C8/hWeYvX+j+A+/5NbpPcdhGQn7vfABngArEfY5tP87//moejx0Ci+om/erdXfD9m/D5hhEh4A+GyOlJUiREQgjMYBWmQjprX/Evy13wJyEHHdP/00PrFh/tJO/1EeAM2EAMjiAGcyz2UCIAJcP2S4Nuap/3aTyWFAUp7ZHGKoDi8ZHy47wMpI36wuslSYgkAcIAN6AAhqHxXyIa+93vpBx5iCAIOCAEDYIG6r3vKYYEScO4JkAAJAAgHBgYSLGjQAAAHGwzIGPjIlTR5/v4drGjxIsaMGjdy7OjxI8iQIkeSLGnyJMqUKleybOnyJcyYFa8ECFARgEGcBwU4mGBhhAwhTbg4smSrW7t5BugNpMdUJtSoUqdSrWr1KtasWrdyfWlDyFchBpo44cLlDJ48lCx5MmCLoLp9A/9R/OfPX9e8evfy7ev3L+DAgqHepWtgor8Ufv0OM57beDDkyJInU65s+fLkgAAAOw==" class="card-img-top figure-img" alt="A Figure">
|
||
<figcaption class="figure-caption"><div class="markdown"><p>The first triangle has area \(1/2\), the second has area \(1/8\), then \(2\) have area \((1/8)^2\), \(4\) have area \((1/8)^3\), ... With some algebra, the total area then should be \(1/2 \cdot (1 + (1/4) + (1/4)^2 + \cdots) = 2/3\).</p>
|
||
</div> </figcaption>
|
||
</figure>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>The figure illustrates a means to compute the area bounded by the parabola, the line <span class="math inline">\(y=1\)</span> and the line <span class="math inline">\(x=0\)</span> using triangles. It suggests that this area can be found by adding the following sum</p>
|
||
<p><span class="math display">\[
|
||
A = 1/2 + 1/8 + 2 \cdot (1/8)^2 + 4 \cdot (1/8)^3 + \cdots
|
||
\]</span></p>
|
||
<p>This value is <span class="math inline">\(2/3\)</span>, so the area under the curve would be <span class="math inline">\(1/3\)</span>. Forget about this specific value - which through more modern machinery becomes uneventful - and focus for a minute on the method: a problem is solved by a suggestion of an infinite process, in this case the creation of more triangles to approximate the unaccounted for area. This is the so-call method of <a href="http://en.wikipedia.org/wiki/Method_of_exhaustion">exhaustion</a> known since the 5th century BC.</p>
|
||
<p>Archimedes used this method to solve a wide range of area problems related to basic geometric shapes, including a more general statement of what we described above.</p>
|
||
<p>The <span class="math inline">\(\cdots\)</span> in the sum expression are the indication that this process continues and that the answer is at the end of an <em>infinite</em> process. To make this line of reasoning rigorous requires the concept of a limit. The concept of a limit is then an old one, but it wasn’t until the age of calculus that it was formalized.</p>
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||
<p>Next, we illustrate how Archimedes approximated <span class="math inline">\(\pi\)</span> – the ratio of the circumference of a circle to its diameter – using interior and exterior <span class="math inline">\(n\)</span>-gons whose perimeters could be computed.</p>
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<div class="cell" data-hold="true" data-execution_count="5">
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<div class="cell-output cell-output-display" data-execution_count="6">
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<div class="d-flex justify-content-center"> <figure class="figure"> <img 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" class="card-img-top figure-img" alt="A Figure">
|
||
<figcaption class="figure-caption"><div class="markdown"><p>The ratio of the circumference of a circle to its diameter, \(\pi\), can be approximated from above and below by computing the perimeters of the inscribed \(n\)-gons. Archimedes computed the perimeters for \(n\) being \(12\), \(24\), \(48\), and \(96\) to determine that \(3~1/7 \leq \pi \leq 3~10/71\).</p>
|
||
</div> </figcaption>
|
||
</figure>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Here Archimedes uses <em>bounds</em> to constrain an unknown value. Had he been able to compute these bounds for larger and larger <span class="math inline">\(n\)</span> the value of <span class="math inline">\(\pi\)</span> could be more accurately determined. In a “limit” it would be squeezed in to have a specific value, which we now know is an irrational number.</p>
|
||
<p>Continuing these concepts, <a href="http://en.wikipedia.org/wiki/Adequality">Fermat</a> in the 1600s essentially took a limit to find the slope of a tangent line to a polynomial curve. Newton in the late 1600s, exploited the idea in his development of calculus (as did Leibniz). Yet it wasn’t until the 1800s that <a href="http://en.wikipedia.org/wiki/Limit_of_a_function#History">Bolzano</a>, Cauchy and Weierstrass put the idea on a firm footing.</p>
|
||
<p>To make things more precise, we begin by discussing the limit of a univariate function as <span class="math inline">\(x\)</span> approaches <span class="math inline">\(c\)</span>.</p>
|
||
<p>Informally, if a limit exists it is the value that <span class="math inline">\(f(x)\)</span> gets close to as <span class="math inline">\(x\)</span> gets close to - but not equal to - <span class="math inline">\(c\)</span>.</p>
|
||
<p>The modern formulation is due to Weirstrass:</p>
|
||
<blockquote class="blockquote">
|
||
<p>The limit of <span class="math inline">\(f(x)\)</span> as <span class="math inline">\(x\)</span> approaches <span class="math inline">\(c\)</span> is <span class="math inline">\(L\)</span> if for every real <span class="math inline">\(\epsilon > 0\)</span>, there exists a real <span class="math inline">\(\delta > 0\)</span> such that for all real <span class="math inline">\(x\)</span>, <span class="math inline">\(0 < \lvert x − c \rvert < \delta\)</span> implies <span class="math inline">\(\lvert f(x) − L \rvert < \epsilon\)</span>. The notation used is <span class="math inline">\(\lim_{x \rightarrow c}f(x) = L\)</span>.</p>
|
||
</blockquote>
|
||
<p>We comment on this later.</p>
|
||
<p>Cauchy begins his incredibly influential <a href="http://gallica.bnf.fr/ark:/12148/bpt6k90196z/f17.image">treatise</a> on calculus considering two examples, the limit as <span class="math inline">\(x\)</span> goes to <span class="math inline">\(0\)</span> of</p>
|
||
<p><span class="math display">\[
|
||
\frac{\sin(x)}{x} \quad\text{and}\quad (1 + x)^{1/x}.
|
||
\]</span></p>
|
||
<p>These take the indeterminate forms <span class="math inline">\(0/0\)</span> and <span class="math inline">\(1^\infty\)</span>, which are found by just putting <span class="math inline">\(0\)</span> in for <span class="math inline">\(x\)</span>. An expression does not need to be defined at <span class="math inline">\(c\)</span>, as these two aren’t at <span class="math inline">\(c=0\)</span>, to discuss its limit. Cauchy illustrates two methods to approach the questions above. The first is to pull out an inequality:</p>
|
||
<p><span class="math display">\[
|
||
\frac{\sin(x)}{\sin(x)} > \frac{\sin(x)}{x} > \frac{\sin(x)}{\tan(x)}
|
||
\]</span></p>
|
||
<p>which is equivalent to:</p>
|
||
<p><span class="math display">\[
|
||
1 > \frac{\sin(x)}{x} > \cos(x)
|
||
\]</span></p>
|
||
<p>This bounds the expression <span class="math inline">\(\sin(x)/x\)</span> between <span class="math inline">\(1\)</span> and <span class="math inline">\(\cos(x)\)</span> and as <span class="math inline">\(x\)</span> gets close to <span class="math inline">\(0\)</span>, the value of <span class="math inline">\(\cos(x)\)</span> “clearly” goes to <span class="math inline">\(1\)</span>, hence <span class="math inline">\(L\)</span> must be <span class="math inline">\(1\)</span>. This is an application of the squeeze theorem, the same idea Archimedes implied when bounding the value for <span class="math inline">\(\pi\)</span> above and below.</p>
|
||
<p>The above bound comes from this figure, for small <span class="math inline">\(x > 0\)</span>:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="6">
|
||
<div class="cell-output cell-output-display" data-execution_count="7">
|
||
<div class="d-flex justify-content-center"> <figure class="figure"> <img 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" class="card-img-top figure-img" alt="A Figure">
|
||
<figcaption class="figure-caption"><div class="markdown"><p>Triangle ABD has less area than the shaded wedge, which has less area than triangle ACD. Their respective areas are \((1/2)\sin(\theta)\), \((1/2)\theta\), and \((1/2)\tan(\theta)\). The inequality used to show \(\sin(x)/x\) is bounded below by \(\cos(x)\) and above by \(1\) comes from a division by \((1/2) \sin(x)\) and taking reciprocals.</p>
|
||
</div> </figcaption>
|
||
</figure>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>To discuss the case of <span class="math inline">\((1+x)^{1/x}\)</span> it proved convenient to assume <span class="math inline">\(x = 1/m\)</span> for integer values of <span class="math inline">\(m\)</span>. At the time of Cauchy, log tables were available to identify the approximate value of the limit. Cauchy computed the following value from logarithm tables.</p>
|
||
<div class="cell" data-hold="true" data-execution_count="7">
|
||
<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a>x <span class="op">=</span> <span class="fl">1</span><span class="op">/</span><span class="fl">10000</span></span>
|
||
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a>(<span class="fl">1</span> <span class="op">+</span> x)<span class="op">^</span>(<span class="fl">1</span><span class="op">/</span>x)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="8">
|
||
<pre><code>2.7181459268249255</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>A table can show the progression to this value:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="8">
|
||
<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="fu">f</span>(x) <span class="op">=</span> (<span class="fl">1</span> <span class="op">+</span> x)<span class="op">^</span>(<span class="fl">1</span><span class="op">/</span>x)</span>
|
||
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a>xs <span class="op">=</span> [<span class="fl">1</span><span class="op">/</span><span class="fl">10</span><span class="op">^</span>i for i <span class="kw">in</span> <span class="fl">1</span><span class="op">:</span><span class="fl">5</span>]</span>
|
||
<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a>[xs <span class="fu">f</span>.(xs)]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="9">
|
||
<pre><code>5×2 Matrix{Float64}:
|
||
0.1 2.59374
|
||
0.01 2.70481
|
||
0.001 2.71692
|
||
0.0001 2.71815
|
||
1.0e-5 2.71827</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>This progression can be seen to be increasing. Cauchy, in his treatise, can see this through:</p>
|
||
<p><span class="math display">\[
|
||
\begin{align*}
|
||
(1 + \frac{1}{m})^n &= 1 + \frac{1}{1} + \frac{1}{1\cdot 2}(1 = \frac{1}{m}) + \\
|
||
& \frac{1}{1\cdot 2\cdot 3}(1 - \frac{1}{m})(1 - \frac{2}{m}) + \cdots \\
|
||
&+
|
||
\frac{1}{1 \cdot 2 \cdot \cdots \cdot m}(1 - \frac{1}{m}) \cdot \cdots \cdot (1 - \frac{m-1}{m}).
|
||
\end{align*}
|
||
\]</span></p>
|
||
<p>These values are clearly increasing as <span class="math inline">\(m\)</span> increases. Cauchy showed the value was bounded between <span class="math inline">\(2\)</span> and <span class="math inline">\(3\)</span> and had the approximate value above. Then he showed the restriction to integers was not necessary. Later we will use this definition for the exponential function:</p>
|
||
<p><span class="math display">\[
|
||
e^x = \lim_{n \rightarrow \infty} (1 + \frac{x}{n})^n,
|
||
\]</span></p>
|
||
<p>with a suitably defined limit.</p>
|
||
<p>These two cases illustrate that though the definition of the limit exists, the computation of a limit is generally found by other means and the intuition of the value of the limit can be gained numerically.</p>
|
||
<section id="indeterminate-forms" class="level3" data-number="18.0.1">
|
||
<h3 data-number="18.0.1" class="anchored" data-anchor-id="indeterminate-forms"><span class="header-section-number">18.0.1</span> Indeterminate forms</h3>
|
||
<p>First it should be noted that for most of the functions encountered, the concepts of a limit at a typical point <span class="math inline">\(c\)</span> is nothing more than just function evaluation at <span class="math inline">\(c\)</span>. This is because, at a typical point, the functions are nicely behaved (what we will soon call “<em>continuous</em>”). However, most questions asked about limits find points that are not typical. For these, the result of evaluating the function at <span class="math inline">\(c\)</span> is typically undefined, and the value comes in one of several <em>indeterminate forms</em>: <span class="math inline">\(0/0\)</span>, <span class="math inline">\(\infty/\infty\)</span>, <span class="math inline">\(0 \cdot \infty\)</span>, <span class="math inline">\(\infty - \infty\)</span>, <span class="math inline">\(0^0\)</span>, <span class="math inline">\(1^\infty\)</span>, and <span class="math inline">\(\infty^0\)</span>.</p>
|
||
<p><code>Julia</code> can help - at times - identify these indeterminate forms, as many such operations produce <code>NaN</code>. For example:</p>
|
||
<div class="cell" data-execution_count="9">
|
||
<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="fl">0</span><span class="op">/</span><span class="fl">0</span>, <span class="cn">Inf</span><span class="op">/</span><span class="cn">Inf</span>, <span class="fl">0</span> <span class="op">*</span> <span class="cn">Inf</span>, <span class="cn">Inf</span> <span class="op">-</span> <span class="cn">Inf</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="10">
|
||
<pre><code>(NaN, NaN, NaN, NaN)</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>However, the values with powers generally do not help, as the IEEE standard has <code>0^0</code> evaluating to 1:</p>
|
||
<div class="cell" data-execution_count="10">
|
||
<div class="sourceCode cell-code" id="cb8"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a><span class="fl">0</span><span class="op">^</span><span class="fl">0</span>, <span class="fl">1</span><span class="op">^</span><span class="cn">Inf</span>, <span class="cn">Inf</span><span class="op">^</span><span class="fl">0</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="11">
|
||
<pre><code>(1, 1.0, 1.0)</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>However, this can be unreliable, as floating point issues may mask the true evaluation. However, as a cheap trick it can work. So, the limit as <span class="math inline">\(x\)</span> goes to <span class="math inline">\(1\)</span> of <span class="math inline">\(\sin(x)/x\)</span> is simply found by evaluation:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="11">
|
||
<div class="sourceCode cell-code" id="cb10"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a>x <span class="op">=</span> <span class="fl">1</span></span>
|
||
<span id="cb10-2"><a href="#cb10-2" aria-hidden="true" tabindex="-1"></a><span class="fu">sin</span>(x) <span class="op">/</span> x</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="12">
|
||
<pre><code>0.8414709848078965</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>But at <span class="math inline">\(x=0\)</span> we get an indicator that there is an issue with just evaluating the function:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="12">
|
||
<div class="sourceCode cell-code" id="cb12"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb12-1"><a href="#cb12-1" aria-hidden="true" tabindex="-1"></a>x <span class="op">=</span> <span class="fl">0</span></span>
|
||
<span id="cb12-2"><a href="#cb12-2" aria-hidden="true" tabindex="-1"></a><span class="fu">sin</span>(x) <span class="op">/</span> x</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="13">
|
||
<pre><code>NaN</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>The above is really just a heuristic. For some functions this is just not true. For example, the <span class="math inline">\(f(x) = \sqrt{x}\)</span> is only defined on <span class="math inline">\([0, \infty)\)</span> There is technically no limit at <span class="math inline">\(0\)</span>, per se, as the function is not defined around <span class="math inline">\(0\)</span>. Other functions jump at values, and will not have a limit, despite having well defined values. The <code>floor</code> function is the function that rounds down to the nearest integer. At integer values there will be a jump (and hence have no limit), even though the function is defined.</p>
|
||
</section>
|
||
<section id="graphical-approaches-to-limits" class="level2" data-number="18.1">
|
||
<h2 data-number="18.1" class="anchored" data-anchor-id="graphical-approaches-to-limits"><span class="header-section-number">18.1</span> Graphical approaches to limits</h2>
|
||
<p>Let’s return to the function <span class="math inline">\(f(x) = \sin(x)/x\)</span>. This function was studied by Euler as part of his solution to the <a href="http://en.wikipedia.org/wiki/Basel_problem">Basel</a> problem. He knew that near <span class="math inline">\(0\)</span>, <span class="math inline">\(\sin(x) \approx x\)</span>, so the ratio is close to <span class="math inline">\(1\)</span> if <span class="math inline">\(x\)</span> is near <span class="math inline">\(0\)</span>. Hence, the intuition is <span class="math inline">\(\lim_{x \rightarrow 0} \sin(x)/x = 1\)</span>, as Cauchy wrote. We can verify this limit graphically two ways. First, a simple graph shows no issue at <span class="math inline">\(0\)</span>:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="13">
|
||
<div class="sourceCode cell-code" id="cb14"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb14-1"><a href="#cb14-1" aria-hidden="true" tabindex="-1"></a><span class="fu">f</span>(x) <span class="op">=</span> <span class="fu">sin</span>(x)<span class="op">/</span>x</span>
|
||
<span id="cb14-2"><a href="#cb14-2" aria-hidden="true" tabindex="-1"></a>xs, ys <span class="op">=</span> <span class="fu">unzip</span>(f, <span class="op">-</span><span class="cn">pi</span><span class="op">/</span><span class="fl">2</span>, <span class="cn">pi</span><span class="op">/</span><span class="fl">2</span>) <span class="co"># get points used to plot `f`</span></span>
|
||
<span id="cb14-3"><a href="#cb14-3" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(xs, ys)</span>
|
||
<span id="cb14-4"><a href="#cb14-4" aria-hidden="true" tabindex="-1"></a><span class="fu">scatter!</span>(xs, ys)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="14">
|
||
<p><img src="limits_files/figure-html/cell-14-output-1.svg" class="img-fluid"></p>
|
||
</div>
|
||
</div>
|
||
<p>The <span class="math inline">\(y\)</span> values of the graph seem to go to <span class="math inline">\(1\)</span> as the <span class="math inline">\(x\)</span> values get close to <span class="math inline">\(0\)</span>. (That the graph looks defined at <span class="math inline">\(0\)</span> is due to the fact that the points sampled to graph do not include <span class="math inline">\(0\)</span>, as shown through the <code>scatter!</code> command – which can be checked via <code>minimum(abs, xs)</code>.)</p>
|
||
<p>We can also verify Euler’s intuition through this graph:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="14">
|
||
<div class="sourceCode cell-code" id="cb15"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(sin, <span class="op">-</span><span class="cn">pi</span><span class="op">/</span><span class="fl">2</span>, <span class="cn">pi</span><span class="op">/</span><span class="fl">2</span>)</span>
|
||
<span id="cb15-2"><a href="#cb15-2" aria-hidden="true" tabindex="-1"></a><span class="fu">plot!</span>(identity) <span class="co"># the function y = x, like how zero is y = 0</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="15">
|
||
<p><img src="limits_files/figure-html/cell-15-output-1.svg" class="img-fluid"></p>
|
||
</div>
|
||
</div>
|
||
<p>That the two are indistinguishable near <span class="math inline">\(0\)</span> makes it easy to see that their ratio should be going towards <span class="math inline">\(1\)</span>.</p>
|
||
<p>A parametric plot shows the same, we see below the slope at <span class="math inline">\((0,0)\)</span> is <em>basically</em> <span class="math inline">\(1\)</span>, because the two functions are varying at the same rate when they are each near <span class="math inline">\(0\)</span></p>
|
||
<div class="cell" data-hold="true" data-execution_count="15">
|
||
<div class="sourceCode cell-code" id="cb16"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb16-1"><a href="#cb16-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(sin, identity, <span class="op">-</span><span class="cn">pi</span><span class="op">/</span><span class="fl">2</span>, <span class="cn">pi</span><span class="op">/</span><span class="fl">2</span>) <span class="co"># parametric plot</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="16">
|
||
<p><img src="limits_files/figure-html/cell-16-output-1.svg" class="img-fluid"></p>
|
||
</div>
|
||
</div>
|
||
<p>The graphical approach to limits - plotting <span class="math inline">\(f(x)\)</span> around <span class="math inline">\(c\)</span> and observing if the <span class="math inline">\(y\)</span> values seem to converge to an <span class="math inline">\(L\)</span> value when <span class="math inline">\(x\)</span> get close to <span class="math inline">\(c\)</span> - allows us to gather quickly if a function seems to have a limit at <span class="math inline">\(c\)</span>, though the precise value of <span class="math inline">\(L\)</span> may be hard to identify.</p>
|
||
<section id="example" class="level5">
|
||
<h5 class="anchored" data-anchor-id="example">Example</h5>
|
||
<p>This example illustrates the same limit a different way. Sliding the <span class="math inline">\(x\)</span> value towards <span class="math inline">\(0\)</span> shows <span class="math inline">\(f(x) = \sin(x)/x\)</span> approaches a value of <span class="math inline">\(1\)</span>.</p>
|
||
<div id="jsxgraph" style="width: 500px; height: 500px;"></div>
|
||
<div class="cell">
|
||
<div class="sourceCode cell-code hidden" id="cb17" data-startfrom="395" data-source-offset="-1"><pre class="sourceCode js code-with-copy"><code class="sourceCode javascript" style="counter-reset: source-line 394;"><span id="cb17-395"><a href="#cb17-395" aria-hidden="true" tabindex="-1"></a>JXG <span class="op">=</span> <span class="pp">require</span>(<span class="st">"jsxgraph"</span>)</span>
|
||
<span id="cb17-396"><a href="#cb17-396" aria-hidden="true" tabindex="-1"></a></span>
|
||
<span id="cb17-397"><a href="#cb17-397" aria-hidden="true" tabindex="-1"></a>b <span class="op">=</span> JXG<span class="op">.</span><span class="at">JSXGraph</span><span class="op">.</span><span class="fu">initBoard</span>(<span class="st">'jsxgraph'</span><span class="op">,</span> {</span>
|
||
<span id="cb17-398"><a href="#cb17-398" aria-hidden="true" tabindex="-1"></a> <span class="dt">boundingbox</span><span class="op">:</span> [<span class="op">-</span><span class="dv">6</span><span class="op">,</span> <span class="fl">1.2</span><span class="op">,</span> <span class="dv">6</span><span class="op">,-</span><span class="fl">1.2</span>]<span class="op">,</span> <span class="dt">axis</span><span class="op">:</span><span class="kw">true</span></span>
|
||
<span id="cb17-399"><a href="#cb17-399" aria-hidden="true" tabindex="-1"></a>})<span class="op">;</span></span>
|
||
<span id="cb17-400"><a href="#cb17-400" aria-hidden="true" tabindex="-1"></a></span>
|
||
<span id="cb17-401"><a href="#cb17-401" aria-hidden="true" tabindex="-1"></a>f <span class="op">=</span> <span class="kw">function</span>(x) {<span class="cf">return</span> <span class="bu">Math</span><span class="op">.</span><span class="fu">sin</span>(x) <span class="op">/</span> x<span class="op">;</span>}<span class="op">;</span></span>
|
||
<span id="cb17-402"><a href="#cb17-402" aria-hidden="true" tabindex="-1"></a>graph <span class="op">=</span> b<span class="op">.</span><span class="fu">create</span>(<span class="st">"functiongraph"</span><span class="op">,</span> [f<span class="op">,</span> <span class="op">-</span><span class="dv">6</span><span class="op">,</span> <span class="dv">6</span>])</span>
|
||
<span id="cb17-403"><a href="#cb17-403" aria-hidden="true" tabindex="-1"></a>seg <span class="op">=</span> b<span class="op">.</span><span class="fu">create</span>(<span class="st">"line"</span><span class="op">,</span> [[<span class="op">-</span><span class="dv">6</span><span class="op">,</span><span class="dv">0</span>]<span class="op">,</span> [<span class="dv">6</span><span class="op">,</span><span class="dv">0</span>]]<span class="op">,</span> {<span class="dt">fixed</span><span class="op">:</span><span class="kw">true</span>})<span class="op">;</span></span>
|
||
<span id="cb17-404"><a href="#cb17-404" aria-hidden="true" tabindex="-1"></a></span>
|
||
<span id="cb17-405"><a href="#cb17-405" aria-hidden="true" tabindex="-1"></a>X <span class="op">=</span> b<span class="op">.</span><span class="fu">create</span>(<span class="st">"glider"</span><span class="op">,</span> [<span class="dv">2</span><span class="op">,</span> <span class="dv">0</span><span class="op">,</span> seg]<span class="op">,</span> {<span class="dt">name</span><span class="op">:</span><span class="st">"x"</span><span class="op">,</span> <span class="dt">size</span><span class="op">:</span><span class="dv">4</span>})<span class="op">;</span></span>
|
||
<span id="cb17-406"><a href="#cb17-406" aria-hidden="true" tabindex="-1"></a>P <span class="op">=</span> b<span class="op">.</span><span class="fu">create</span>(<span class="st">"point"</span><span class="op">,</span> [<span class="kw">function</span>() {<span class="cf">return</span> X<span class="op">.</span><span class="fu">X</span>()}<span class="op">,</span> <span class="kw">function</span>() {<span class="cf">return</span> <span class="fu">f</span>(X<span class="op">.</span><span class="fu">X</span>())}]<span class="op">,</span> {<span class="dt">name</span><span class="op">:</span><span class="st">""</span>})<span class="op">;</span></span>
|
||
<span id="cb17-407"><a href="#cb17-407" aria-hidden="true" tabindex="-1"></a>Q <span class="op">=</span> b<span class="op">.</span><span class="fu">create</span>(<span class="st">"point"</span><span class="op">,</span> [<span class="dv">0</span><span class="op">,</span> <span class="kw">function</span>() {<span class="cf">return</span> P<span class="op">.</span><span class="fu">Y</span>()<span class="op">;</span>}]<span class="op">,</span> {<span class="dt">name</span><span class="op">:</span><span class="st">"f(x)"</span>})<span class="op">;</span></span>
|
||
<span id="cb17-408"><a href="#cb17-408" aria-hidden="true" tabindex="-1"></a></span>
|
||
<span id="cb17-409"><a href="#cb17-409" aria-hidden="true" tabindex="-1"></a>segup <span class="op">=</span> b<span class="op">.</span><span class="fu">create</span>(<span class="st">"segment"</span><span class="op">,</span> [P<span class="op">,</span>X]<span class="op">,</span> {<span class="dt">dash</span><span class="op">:</span><span class="dv">2</span>})<span class="op">;</span></span>
|
||
<span id="cb17-410"><a href="#cb17-410" aria-hidden="true" tabindex="-1"></a>segover <span class="op">=</span> b<span class="op">.</span><span class="fu">create</span>(<span class="st">"segment"</span><span class="op">,</span> [P<span class="op">,</span> [<span class="dv">0</span><span class="op">,</span> <span class="kw">function</span>() {<span class="cf">return</span> P<span class="op">.</span><span class="fu">Y</span>()}]]<span class="op">,</span> {<span class="dt">dash</span><span class="op">:</span><span class="dv">2</span>})<span class="op">;</span></span>
|
||
<span id="cb17-411"><a href="#cb17-411" aria-hidden="true" tabindex="-1"></a></span>
|
||
<span id="cb17-412"><a href="#cb17-412" aria-hidden="true" tabindex="-1"></a></span>
|
||
<span id="cb17-413"><a href="#cb17-413" aria-hidden="true" tabindex="-1"></a>txt <span class="op">=</span> b<span class="op">.</span><span class="fu">create</span>(<span class="st">'text'</span><span class="op">,</span> [<span class="dv">2</span><span class="op">,</span> <span class="dv">1</span><span class="op">,</span> <span class="kw">function</span>() {</span>
|
||
<span id="cb17-414"><a href="#cb17-414" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> <span class="st">"x = "</span> <span class="op">+</span> X<span class="op">.</span><span class="fu">X</span>()<span class="op">.</span><span class="fu">toFixed</span>(<span class="dv">4</span>) <span class="op">+</span> <span class="st">", f(x) = "</span> <span class="op">+</span> P<span class="op">.</span><span class="fu">Y</span>()<span class="op">.</span><span class="fu">toFixed</span>(<span class="dv">4</span>)<span class="op">;</span></span>
|
||
<span id="cb17-415"><a href="#cb17-415" aria-hidden="true" tabindex="-1"></a>}])<span class="op">;</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display hidden">
|
||
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|
||
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|
||
|
||
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|
||
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|
||
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|
||
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|
||
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|
||
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|
||
|
||
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|
||
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|
||
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|
||
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|
||
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|
||
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|
||
|
||
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|
||
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|
||
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|
||
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|
||
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|
||
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|
||
|
||
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|
||
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|
||
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|
||
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|
||
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|
||
<div id="ojs-cell-1-5" data-nodetype="declaration">
|
||
|
||
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|
||
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|
||
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|
||
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|
||
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|
||
<div id="ojs-cell-1-6" data-nodetype="declaration">
|
||
|
||
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|
||
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|
||
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|
||
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|
||
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|
||
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|
||
|
||
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|
||
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|
||
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|
||
<div class="cell-output cell-output-display hidden">
|
||
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|
||
<div id="ojs-cell-1-8" data-nodetype="declaration">
|
||
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell-output cell-output-display hidden">
|
||
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|
||
<div id="ojs-cell-1-9" data-nodetype="declaration">
|
||
|
||
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|
||
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|
||
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|
||
<div class="cell-output cell-output-display hidden">
|
||
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|
||
<div id="ojs-cell-1-10" data-nodetype="declaration">
|
||
|
||
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|
||
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|
||
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|
||
<div class="cell-output cell-output-display hidden">
|
||
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|
||
<div id="ojs-cell-1-11" data-nodetype="declaration">
|
||
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="example-1" class="level5">
|
||
<h5 class="anchored" data-anchor-id="example-1">Example</h5>
|
||
<p>Consider now the following limit</p>
|
||
<p><span class="math display">\[
|
||
\lim_{x \rightarrow 2} \frac{x^2 - 5x + 6}{x^2 +x - 6}
|
||
\]</span></p>
|
||
<p>Noting that this is a ratio of nice polynomial functions, we first check whether there is anything to do:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="16">
|
||
<div class="sourceCode cell-code" id="cb18"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb18-1"><a href="#cb18-1" aria-hidden="true" tabindex="-1"></a><span class="fu">f</span>(x) <span class="op">=</span> (x<span class="op">^</span><span class="fl">2</span> <span class="op">-</span> <span class="fl">5</span>x <span class="op">+</span> <span class="fl">6</span>) <span class="op">/</span> (x<span class="op">^</span><span class="fl">2</span> <span class="op">+</span> x <span class="op">-</span> <span class="fl">6</span>)</span>
|
||
<span id="cb18-2"><a href="#cb18-2" aria-hidden="true" tabindex="-1"></a>c <span class="op">=</span> <span class="fl">2</span></span>
|
||
<span id="cb18-3"><a href="#cb18-3" aria-hidden="true" tabindex="-1"></a><span class="fu">f</span>(c)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="17">
|
||
<pre><code>NaN</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>The <code>NaN</code> indicates that this function is indeterminate at <span class="math inline">\(c=2\)</span>. A quick plot gives us an idea that the limit exists and is roughly <span class="math inline">\(-0.2\)</span>:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="17">
|
||
<div class="sourceCode cell-code" id="cb20"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb20-1"><a href="#cb20-1" aria-hidden="true" tabindex="-1"></a>c, delta <span class="op">=</span> <span class="fl">2</span>, <span class="fl">1</span></span>
|
||
<span id="cb20-2"><a href="#cb20-2" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(x <span class="op">-></span> (x<span class="op">^</span><span class="fl">2</span> <span class="op">-</span> <span class="fl">5</span>x <span class="op">+</span> <span class="fl">6</span>) <span class="op">/</span> (x<span class="op">^</span><span class="fl">2</span> <span class="op">+</span> x <span class="op">-</span> <span class="fl">6</span>), c <span class="op">-</span> delta, c <span class="op">+</span> delta)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="18">
|
||
<p><img src="limits_files/figure-html/cell-18-output-1.svg" class="img-fluid"></p>
|
||
</div>
|
||
</div>
|
||
<p>The graph looks “continuous.” In fact, the value <span class="math inline">\(c=2\)</span> is termed a <em>removable singularity</em> as redefining <span class="math inline">\(f(x)\)</span> to be <span class="math inline">\(-0.2\)</span> when <span class="math inline">\(x=2\)</span> results in a “continuous” function.</p>
|
||
<p>As an aside, we can redefine <code>f</code> using the “ternary operator”:</p>
|
||
<div class="sourceCode cell-code" id="cb21"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb21-1"><a href="#cb21-1" aria-hidden="true" tabindex="-1"></a><span class="fu">f</span>(x) <span class="op">=</span> x <span class="op">==</span> <span class="fl">2.0</span> ? <span class="op">-</span><span class="fl">0.2</span> <span class="op">:</span> (x<span class="op">^</span><span class="fl">2</span> <span class="op">-</span> <span class="fl">5</span>x <span class="op">+</span> <span class="fl">6</span>) <span class="op">/</span> (x<span class="op">^</span><span class="fl">2</span> <span class="op">+</span> x <span class="op">-</span> <span class="fl">6</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<p>This particular case is a textbook example: one can easily factor <span class="math inline">\(f(x)\)</span> to get:</p>
|
||
<p><span class="math display">\[
|
||
f(x) = \frac{(x-2)(x-3)}{(x-2)(x+3)}
|
||
\]</span></p>
|
||
<p>Written in this form, we clearly see that this is the same function as <span class="math inline">\(g(x) = (x-3)/(x+3)\)</span> when <span class="math inline">\(x \neq 2\)</span>. The function <span class="math inline">\(g(x)\)</span> is “continuous” at <span class="math inline">\(x=2\)</span>. So were one to redefine <span class="math inline">\(f(x)\)</span> at <span class="math inline">\(x=2\)</span> to be <span class="math inline">\(g(2) = (2 - 3)/(2 + 3) = -0.2\)</span> it would be made continuous, hence the term removable singularity.</p>
|
||
</section>
|
||
</section>
|
||
<section id="numerical-approaches-to-limits" class="level2" data-number="18.2">
|
||
<h2 data-number="18.2" class="anchored" data-anchor-id="numerical-approaches-to-limits"><span class="header-section-number">18.2</span> Numerical approaches to limits</h2>
|
||
<p>The investigation of <span class="math inline">\(\lim_{x \rightarrow 0}(1 + x)^{1/x}\)</span> by evaluating the function at <span class="math inline">\(1/10000\)</span> by Cauchy can be done much more easily nowadays. As does a graphical approach, a numerical approach can give insight into a limit and often a good numeric estimate.</p>
|
||
<p>The basic idea is to create a sequence of <span class="math inline">\(x\)</span> values going towards <span class="math inline">\(c\)</span> and then investigate if the corresponding <span class="math inline">\(y\)</span> values are eventually near some <span class="math inline">\(L\)</span>.</p>
|
||
<p>Best, to see by example. Suppose we are asked to investigate</p>
|
||
<p><span class="math display">\[
|
||
\lim_{x \rightarrow 25} \frac{\sqrt{x} - 5}{\sqrt{x - 16} - 3}.
|
||
\]</span></p>
|
||
<p>We first define a function and check if there are issues at <span class="math inline">\(25\)</span>:</p>
|
||
<div class="cell" data-execution_count="19">
|
||
<div class="sourceCode cell-code" id="cb22"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb22-1"><a href="#cb22-1" aria-hidden="true" tabindex="-1"></a><span class="fu">f</span>(x) <span class="op">=</span> (<span class="fu">sqrt</span>(x) <span class="op">-</span> <span class="fl">5</span>) <span class="op">/</span> (<span class="fu">sqrt</span>(x<span class="op">-</span><span class="fl">16</span>) <span class="op">-</span> <span class="fl">3</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="19">
|
||
<pre><code>f (generic function with 1 method)</code></pre>
|
||
</div>
|
||
</div>
|
||
<div class="cell" data-execution_count="20">
|
||
<div class="sourceCode cell-code" id="cb24"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb24-1"><a href="#cb24-1" aria-hidden="true" tabindex="-1"></a>c <span class="op">=</span> <span class="fl">25</span></span>
|
||
<span id="cb24-2"><a href="#cb24-2" aria-hidden="true" tabindex="-1"></a><span class="fu">f</span>(c)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="20">
|
||
<pre><code>NaN</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>So yes, an issue of the indeterminate form <span class="math inline">\(0/0\)</span>. We investigate numerically by making a set of numbers getting close to <span class="math inline">\(c\)</span>. This is most easily done making numbers getting close to <span class="math inline">\(0\)</span> and adding them to or subtracting them from <span class="math inline">\(c\)</span>. Some natural candidates are negative powers of <span class="math inline">\(10\)</span>:</p>
|
||
<div class="cell" data-execution_count="21">
|
||
<div class="sourceCode cell-code" id="cb26"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb26-1"><a href="#cb26-1" aria-hidden="true" tabindex="-1"></a>hs <span class="op">=</span> [<span class="fl">1</span><span class="op">/</span><span class="fl">10</span><span class="op">^</span>i for i <span class="kw">in</span> <span class="fl">1</span><span class="op">:</span><span class="fl">8</span>]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="21">
|
||
<pre><code>8-element Vector{Float64}:
|
||
0.1
|
||
0.01
|
||
0.001
|
||
0.0001
|
||
1.0e-5
|
||
1.0e-6
|
||
1.0e-7
|
||
1.0e-8</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>We can add these to <span class="math inline">\(c\)</span> and then evaluate:</p>
|
||
<div class="cell" data-execution_count="22">
|
||
<div class="sourceCode cell-code" id="cb28"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb28-1"><a href="#cb28-1" aria-hidden="true" tabindex="-1"></a>xs <span class="op">=</span> c <span class="op">.+</span> hs</span>
|
||
<span id="cb28-2"><a href="#cb28-2" aria-hidden="true" tabindex="-1"></a>ys <span class="op">=</span> <span class="fu">f</span>.(xs)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="22">
|
||
<pre><code>8-element Vector{Float64}:
|
||
0.6010616008415922
|
||
0.6001066157341047
|
||
0.6000106661569936
|
||
0.6000010666430725
|
||
0.6000001065281493
|
||
0.6000000122568625
|
||
0.5999999946709295
|
||
0.6</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>To visualize, we can put in a table using <code>[xs ys]</code> notation:</p>
|
||
<div class="cell" data-execution_count="23">
|
||
<div class="sourceCode cell-code" id="cb30"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb30-1"><a href="#cb30-1" aria-hidden="true" tabindex="-1"></a>[xs ys]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="23">
|
||
<pre><code>8×2 Matrix{Float64}:
|
||
25.1 0.601062
|
||
25.01 0.600107
|
||
25.001 0.600011
|
||
25.0001 0.600001
|
||
25.0 0.6
|
||
25.0 0.6
|
||
25.0 0.6
|
||
25.0 0.6</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>The <span class="math inline">\(y\)</span>-values seem to be getting near <span class="math inline">\(0.6\)</span>.</p>
|
||
<p>Since limits are defined by the expression <span class="math inline">\(0 < \lvert x-c\rvert < \delta\)</span>, we should also look at values smaller than <span class="math inline">\(c\)</span>. There isn’t much difference (note the <code>.-</code> sign in <code>c .- hs</code>):</p>
|
||
<div class="cell" data-hold="true" data-execution_count="24">
|
||
<div class="sourceCode cell-code" id="cb32"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb32-1"><a href="#cb32-1" aria-hidden="true" tabindex="-1"></a>xs <span class="op">=</span> c <span class="op">.-</span> hs</span>
|
||
<span id="cb32-2"><a href="#cb32-2" aria-hidden="true" tabindex="-1"></a>ys <span class="op">=</span> <span class="fu">f</span>.(xs)</span>
|
||
<span id="cb32-3"><a href="#cb32-3" aria-hidden="true" tabindex="-1"></a>[xs ys]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="24">
|
||
<pre><code>8×2 Matrix{Float64}:
|
||
24.9 0.598928
|
||
24.99 0.599893
|
||
24.999 0.599989
|
||
24.9999 0.599999
|
||
25.0 0.6
|
||
25.0 0.6
|
||
25.0 0.6
|
||
25.0 0.6</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>Same story. The numeric evidence supports a limit of <span class="math inline">\(L=0.6\)</span>.</p>
|
||
<section id="example-the-secant-line" class="level5">
|
||
<h5 class="anchored" data-anchor-id="example-the-secant-line">Example: the secant line</h5>
|
||
<p>Let <span class="math inline">\(f(x) = x^x\)</span> and consider the ratio:</p>
|
||
<p><span class="math display">\[
|
||
\frac{f(c + h) - f(c)}{h}
|
||
\]</span></p>
|
||
<p>As <span class="math inline">\(h\)</span> goes to <span class="math inline">\(0\)</span>, this will take the form <span class="math inline">\(0/0\)</span> in most cases, and in the particular case of <span class="math inline">\(f(x) = x^x\)</span> and <span class="math inline">\(c=1\)</span> it will be. The expression has a geometric interpretation of being the slope of the secant line connecting the two points <span class="math inline">\((c,f(c))\)</span> and <span class="math inline">\((c+h, f(c+h))\)</span>.</p>
|
||
<p>To look at the limit in this example, we have (recycling the values in <code>hs</code>):</p>
|
||
<div class="cell" data-hold="true" data-execution_count="25">
|
||
<div class="sourceCode cell-code" id="cb34"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb34-1"><a href="#cb34-1" aria-hidden="true" tabindex="-1"></a>c <span class="op">=</span> <span class="fl">1</span></span>
|
||
<span id="cb34-2"><a href="#cb34-2" aria-hidden="true" tabindex="-1"></a><span class="fu">f</span>(x) <span class="op">=</span> x<span class="op">^</span>x</span>
|
||
<span id="cb34-3"><a href="#cb34-3" aria-hidden="true" tabindex="-1"></a>ys <span class="op">=</span> [(<span class="fu">f</span>(c <span class="op">+</span> h) <span class="op">-</span> <span class="fu">f</span>(c)) <span class="op">/</span> h for h <span class="kw">in</span> hs]</span>
|
||
<span id="cb34-4"><a href="#cb34-4" aria-hidden="true" tabindex="-1"></a>[hs ys]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="25">
|
||
<pre><code>8×2 Matrix{Float64}:
|
||
0.1 1.10534
|
||
0.01 1.01005
|
||
0.001 1.001
|
||
0.0001 1.0001
|
||
1.0e-5 1.00001
|
||
1.0e-6 1.0
|
||
1.0e-7 1.0
|
||
1.0e-8 1.0</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>The limit looks like <span class="math inline">\(L=1\)</span>. A similar check on the left will confirm this numerically.</p>
|
||
</section>
|
||
<section id="issues-with-the-numeric-approach" class="level3" data-number="18.2.1">
|
||
<h3 data-number="18.2.1" class="anchored" data-anchor-id="issues-with-the-numeric-approach"><span class="header-section-number">18.2.1</span> Issues with the numeric approach</h3>
|
||
<p>The numeric approach often gives a good intuition as to the existence of a limit and its value. However, it can be misleading. Consider this limit question:</p>
|
||
<p><span class="math display">\[
|
||
\lim_{x \rightarrow 0} \frac{1 - \cos(x)}{x^2}.
|
||
\]</span></p>
|
||
<p>We can see that it is indeterminate of the form <span class="math inline">\(0/0\)</span>:</p>
|
||
<div class="cell" data-execution_count="26">
|
||
<div class="sourceCode cell-code" id="cb36"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb36-1"><a href="#cb36-1" aria-hidden="true" tabindex="-1"></a><span class="fu">g</span>(x) <span class="op">=</span> (<span class="fl">1</span> <span class="op">-</span> <span class="fu">cos</span>(x)) <span class="op">/</span> x<span class="op">^</span><span class="fl">2</span></span>
|
||
<span id="cb36-2"><a href="#cb36-2" aria-hidden="true" tabindex="-1"></a><span class="fu">g</span>(<span class="fl">0</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="26">
|
||
<pre><code>NaN</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>What is the value of <span class="math inline">\(L\)</span>, if it exists? A quick attempt numerically yields:</p>
|
||
<div class="cell" data-execution_count="27">
|
||
<div class="sourceCode cell-code" id="cb38"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb38-1"><a href="#cb38-1" aria-hidden="true" tabindex="-1"></a>𝒙s <span class="op">=</span> <span class="fl">0</span> <span class="op">.+</span> hs</span>
|
||
<span id="cb38-2"><a href="#cb38-2" aria-hidden="true" tabindex="-1"></a>𝒚s <span class="op">=</span> [<span class="fu">g</span>(x) for x <span class="kw">in</span> 𝒙s]</span>
|
||
<span id="cb38-3"><a href="#cb38-3" aria-hidden="true" tabindex="-1"></a>[𝒙s 𝒚s]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="27">
|
||
<pre><code>8×2 Matrix{Float64}:
|
||
0.1 0.499583
|
||
0.01 0.499996
|
||
0.001 0.5
|
||
0.0001 0.5
|
||
1.0e-5 0.5
|
||
1.0e-6 0.500044
|
||
1.0e-7 0.4996
|
||
1.0e-8 0.0</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>Hmm, the values in <code>ys</code> appear to be going to <span class="math inline">\(0.5\)</span>, but then end up at <span class="math inline">\(0\)</span>. Is the limit <span class="math inline">\(0\)</span> or <span class="math inline">\(1/2\)</span>? The answer is <span class="math inline">\(1/2\)</span>. The last <span class="math inline">\(0\)</span> is an artifact of floating point arithmetic and the last few deviations from <code>0.5</code> due to loss of precision in subtraction. To investigate, we look more carefully at the two ratios:</p>
|
||
<div class="cell" data-execution_count="28">
|
||
<div class="sourceCode cell-code" id="cb40"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb40-1"><a href="#cb40-1" aria-hidden="true" tabindex="-1"></a>y1s <span class="op">=</span> [<span class="fl">1</span> <span class="op">-</span> <span class="fu">cos</span>(x) for x <span class="kw">in</span> 𝒙s]</span>
|
||
<span id="cb40-2"><a href="#cb40-2" aria-hidden="true" tabindex="-1"></a>y2s <span class="op">=</span> [x<span class="op">^</span><span class="fl">2</span> for x <span class="kw">in</span> 𝒙s]</span>
|
||
<span id="cb40-3"><a href="#cb40-3" aria-hidden="true" tabindex="-1"></a>[𝒙s y1s y2s]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="28">
|
||
<pre><code>8×3 Matrix{Float64}:
|
||
0.1 0.00499583 0.01
|
||
0.01 4.99996e-5 0.0001
|
||
0.001 5.0e-7 1.0e-6
|
||
0.0001 5.0e-9 1.0e-8
|
||
1.0e-5 5.0e-11 1.0e-10
|
||
1.0e-6 5.00044e-13 1.0e-12
|
||
1.0e-7 4.996e-15 1.0e-14
|
||
1.0e-8 0.0 1.0e-16</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>Looking at the bottom of the second column reveals the error. The value of <code>1 - cos(1.0e-8)</code> is <code>0</code> and not a value around <code>5e-17</code>, as would be expected from the pattern above it. This is because the smallest floating point value less than <code>1.0</code> is more than <code>5e-17</code> units away, so <code>cos(1e-8)</code> is evaluated to be <code>1.0</code>. There just isn’t enough granularity to get this close to <span class="math inline">\(0\)</span>.</p>
|
||
<p>Not that we needed to. The answer would have been clear if we had stopped with <code>x=1e-6</code>, say.</p>
|
||
<p>In general, some functions will frustrate the numeric approach. It is best to be wary of results. At a minimum they should confirm what a quick graph shows, though even that isn’t enough, as this next example shows.</p>
|
||
<section id="example-2" class="level5">
|
||
<h5 class="anchored" data-anchor-id="example-2">Example</h5>
|
||
<p>Let <span class="math inline">\(h(x)\)</span> be defined by</p>
|
||
<p><span class="math display">\[
|
||
h(x) = x^2 + 1 + \log(| 11 \cdot x - 15 |)/99.
|
||
\]</span></p>
|
||
<p>The question is to investigate</p>
|
||
<p><span class="math display">\[
|
||
\lim_{x \rightarrow 15/11} h(x)
|
||
\]</span></p>
|
||
<p>A plot shows the answer appears to be straightforward:</p>
|
||
<div class="cell" data-execution_count="29">
|
||
<div class="cell-output cell-output-display" data-execution_count="29">
|
||
<p><img src="limits_files/figure-html/cell-30-output-1.svg" class="img-fluid"></p>
|
||
</div>
|
||
</div>
|
||
<p>Taking values near <span class="math inline">\(15/11\)</span> shows nothing too unusual:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="30">
|
||
<div class="sourceCode cell-code" id="cb42"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb42-1"><a href="#cb42-1" aria-hidden="true" tabindex="-1"></a>c <span class="op">=</span> <span class="fl">15</span><span class="op">/</span><span class="fl">11</span></span>
|
||
<span id="cb42-2"><a href="#cb42-2" aria-hidden="true" tabindex="-1"></a>hs <span class="op">=</span> [<span class="fl">1</span><span class="op">/</span><span class="fl">10</span><span class="op">^</span>i for i <span class="kw">in</span> <span class="fl">4</span><span class="op">:</span><span class="fl">3</span><span class="op">:</span><span class="fl">16</span>]</span>
|
||
<span id="cb42-3"><a href="#cb42-3" aria-hidden="true" tabindex="-1"></a>xs <span class="op">=</span> c <span class="op">.+</span> hs</span>
|
||
<span id="cb42-4"><a href="#cb42-4" aria-hidden="true" tabindex="-1"></a>[xs <span class="fu">h</span>.(xs)]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="30">
|
||
<pre><code>5×2 Matrix{Float64}:
|
||
1.36374 2.79096
|
||
1.36364 2.72092
|
||
1.36364 2.65114
|
||
1.36364 2.58135
|
||
1.36364 2.51643</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>(Though both the graph and the table hint at something a bit odd.)</p>
|
||
<p>However the limit in this case is <span class="math inline">\(-\infty\)</span> (or DNE), as there is an aysmptote at <span class="math inline">\(c=15/11\)</span>. The problem is the asymptote due to the logarithm is extremely narrow and happens between floating point values to the left and right of <span class="math inline">\(15/11\)</span>.</p>
|
||
</section>
|
||
</section>
|
||
<section id="richardson-extrapolation" class="level3" data-number="18.2.2">
|
||
<h3 data-number="18.2.2" class="anchored" data-anchor-id="richardson-extrapolation"><span class="header-section-number">18.2.2</span> Richardson extrapolation</h3>
|
||
<p>The <a href="https://github.com/JuliaMath/Richardson.jl"><code>Richardson</code></a> package provides a function to extrapolate a function <code>f(x)</code> to <code>f(x0)</code>, as the numeric limit does. We illustrate its use by example:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="31">
|
||
<div class="sourceCode cell-code" id="cb44"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb44-1"><a href="#cb44-1" aria-hidden="true" tabindex="-1"></a><span class="fu">f</span>(x) <span class="op">=</span> <span class="fu">sin</span>(x)<span class="op">/</span>x</span>
|
||
<span id="cb44-2"><a href="#cb44-2" aria-hidden="true" tabindex="-1"></a><span class="fu">extrapolate</span>(f, <span class="fl">1</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="31">
|
||
<pre><code>(0.9999999999922424, 4.538478481919128e-9)</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>The answer involves two terms, the second being an estimate for the error in the estimation of <code>f(0)</code>.</p>
|
||
<p>The values the method chooses could be viewed as follows:</p>
|
||
<div class="cell" data-term="true" data-execution_count="32">
|
||
<div class="sourceCode cell-code" id="cb46"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb46-1"><a href="#cb46-1" aria-hidden="true" tabindex="-1"></a><span class="fu">extrapolate</span>(<span class="fl">1</span>) <span class="cf">do</span> x <span class="co"># using `do` notation for the function</span></span>
|
||
<span id="cb46-2"><a href="#cb46-2" aria-hidden="true" tabindex="-1"></a> <span class="pp">@show</span> x</span>
|
||
<span id="cb46-3"><a href="#cb46-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">sin</span>(x)<span class="op">/</span>x</span>
|
||
<span id="cb46-4"><a href="#cb46-4" aria-hidden="true" tabindex="-1"></a><span class="cf">end</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-stdout">
|
||
<pre><code>x = 1.0
|
||
x = 0.125
|
||
x = 0.015625
|
||
x = 0.001953125
|
||
x = 0.000244140625</code></pre>
|
||
</div>
|
||
<div class="cell-output cell-output-display" data-execution_count="32">
|
||
<pre><code>(0.9999999999922424, 4.538478481919128e-9)</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>The <code>extrapolate</code> function avoids the numeric problems encountered in the following example</p>
|
||
<div class="cell" data-hold="true" data-execution_count="33">
|
||
<div class="sourceCode cell-code" id="cb49"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb49-1"><a href="#cb49-1" aria-hidden="true" tabindex="-1"></a><span class="fu">f</span>(x) <span class="op">=</span> (<span class="fl">1</span> <span class="op">-</span> <span class="fu">cos</span>(x)) <span class="op">/</span> x<span class="op">^</span><span class="fl">2</span></span>
|
||
<span id="cb49-2"><a href="#cb49-2" aria-hidden="true" tabindex="-1"></a><span class="fu">extrapolate</span>(f, <span class="fl">1</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="33">
|
||
<pre><code>(0.5000000007193545, 4.705535960880525e-11)</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>To find limits at a value of <code>c</code> not equal to <code>0</code>, we set the <code>x_0</code> argument. For example,</p>
|
||
<div class="cell" data-hold="true" data-execution_count="34">
|
||
<div class="sourceCode cell-code" id="cb51"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb51-1"><a href="#cb51-1" aria-hidden="true" tabindex="-1"></a><span class="fu">f</span>(x) <span class="op">=</span> (<span class="fu">sqrt</span>(x) <span class="op">-</span> <span class="fl">5</span>) <span class="op">/</span> (<span class="fu">sqrt</span>(x<span class="op">-</span><span class="fl">16</span>) <span class="op">-</span> <span class="fl">3</span>)</span>
|
||
<span id="cb51-2"><a href="#cb51-2" aria-hidden="true" tabindex="-1"></a>c <span class="op">=</span> <span class="fl">25</span></span>
|
||
<span id="cb51-3"><a href="#cb51-3" aria-hidden="true" tabindex="-1"></a><span class="fu">extrapolate</span>(f, <span class="fl">26</span>, x0<span class="op">=</span><span class="fl">25</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="34">
|
||
<pre><code>(0.6000000000015944, 4.5619086286308175e-10)</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>This value can also be <code>Inf</code>, in anticipation of infinite limits to be discussed in a subsequent section:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="35">
|
||
<div class="sourceCode cell-code" id="cb53"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb53-1"><a href="#cb53-1" aria-hidden="true" tabindex="-1"></a><span class="fu">f</span>(x) <span class="op">=</span> (x<span class="op">^</span><span class="fl">2</span> <span class="op">-</span> <span class="fl">2</span>x <span class="op">+</span> <span class="fl">1</span>)<span class="op">/</span>(x<span class="op">^</span><span class="fl">3</span> <span class="op">-</span> <span class="fl">3</span>x<span class="op">^</span><span class="fl">2</span> <span class="op">+</span> <span class="fl">2</span>x <span class="op">+</span> <span class="fl">1</span>)</span>
|
||
<span id="cb53-2"><a href="#cb53-2" aria-hidden="true" tabindex="-1"></a><span class="fu">extrapolate</span>(f, <span class="fl">10</span>, x0<span class="op">=</span><span class="cn">Inf</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="35">
|
||
<pre><code>(0.0, 0.0)</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>(The starting value should be to the right of any zeros of the denominator.)</p>
|
||
</section>
|
||
</section>
|
||
<section id="symbolic-approach-to-limits" class="level2" data-number="18.3">
|
||
<h2 data-number="18.3" class="anchored" data-anchor-id="symbolic-approach-to-limits"><span class="header-section-number">18.3</span> Symbolic approach to limits</h2>
|
||
<p>The <code>SymPy</code> package provides a <code>limit</code> function for finding the limit of an expression in a given variable. It must be loaded, as was done initially. The <code>limit</code> function’s use requires the expression, the variable and a value for <span class="math inline">\(c\)</span>. (Similar to the three things in the notation <span class="math inline">\(\lim_{x \rightarrow c}f(x)\)</span>.)</p>
|
||
<p>For example, the limit at <span class="math inline">\(0\)</span> of <span class="math inline">\((1-\cos(x))/x^2\)</span> is easily handled:</p>
|
||
<div class="cell" data-execution_count="36">
|
||
<div class="sourceCode cell-code" id="cb55"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb55-1"><a href="#cb55-1" aria-hidden="true" tabindex="-1"></a><span class="pp">@syms</span> x<span class="op">::</span><span class="dt">real</span></span>
|
||
<span id="cb55-2"><a href="#cb55-2" aria-hidden="true" tabindex="-1"></a><span class="fu">limit</span>((<span class="fl">1</span> <span class="op">-</span> <span class="fu">cos</span>(x)) <span class="op">/</span> x<span class="op">^</span><span class="fl">2</span>, x <span class="op">=></span> <span class="fl">0</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="36">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
\frac{1}{2}
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<p>The pair notation (<code>x => 0</code>) is used to indicate the variable and the value it is going to.</p>
|
||
<section id="example-3" class="level5">
|
||
<h5 class="anchored" data-anchor-id="example-3">Example</h5>
|
||
<p>We look again at this function which despite having a vertical asymptote at <span class="math inline">\(x=15/11\)</span> has the property that it is positive for all floating point values, making both a numeric and graphical approach impossible:</p>
|
||
<p><span class="math display">\[
|
||
f(x) = x^2 + 1 + \log(| 11 \cdot x - 15 |)/99.
|
||
\]</span></p>
|
||
<p>We find the limit symbolically at <span class="math inline">\(c=15/11\)</span> as follows, taking care to use the exact value <code>15//11</code> and not the <em>floating point</em> approximation returned by <code>15/11</code>:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="37">
|
||
<div class="sourceCode cell-code" id="cb56"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb56-1"><a href="#cb56-1" aria-hidden="true" tabindex="-1"></a><span class="fu">f</span>(x) <span class="op">=</span> x<span class="op">^</span><span class="fl">2</span> <span class="op">+</span> <span class="fl">1</span> <span class="op">+</span> <span class="fu">log</span>(<span class="fu">abs</span>(<span class="fl">11</span>x <span class="op">-</span> <span class="fl">15</span>))<span class="op">/</span><span class="fl">99</span></span>
|
||
<span id="cb56-2"><a href="#cb56-2" aria-hidden="true" tabindex="-1"></a><span class="fu">limit</span>(<span class="fu">f</span>(x), x <span class="op">=></span> <span class="fl">15</span> <span class="op">//</span> <span class="fl">11</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="37">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
-\infty
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="example-4" class="level5">
|
||
<h5 class="anchored" data-anchor-id="example-4">Example</h5>
|
||
<p>Find the <a href="http://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule">limits</a>:</p>
|
||
<p><span class="math display">\[
|
||
\lim_{x \rightarrow 0} \frac{2\sin(x) - \sin(2x)}{x - \sin(x)}, \quad
|
||
\lim_{x \rightarrow 0} \frac{e^x - 1 - x}{x^2}, \quad
|
||
\lim_{\rho \rightarrow 0} \frac{x^{1-\rho} - 1}{1 - \rho}.
|
||
\]</span></p>
|
||
<p>We have for the first:</p>
|
||
<div class="cell" data-execution_count="38">
|
||
<div class="sourceCode cell-code" id="cb57"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb57-1"><a href="#cb57-1" aria-hidden="true" tabindex="-1"></a><span class="fu">limit</span>( (<span class="fl">2</span><span class="fu">sin</span>(x) <span class="op">-</span> <span class="fu">sin</span>(<span class="fl">2</span>x)) <span class="op">/</span> (x <span class="op">-</span> <span class="fu">sin</span>(x)), x <span class="op">=></span> <span class="fl">0</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="38">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
6
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<p>The second is similarly done, though here we define a function for variety:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="39">
|
||
<div class="sourceCode cell-code" id="cb58"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb58-1"><a href="#cb58-1" aria-hidden="true" tabindex="-1"></a><span class="fu">f</span>(x) <span class="op">=</span> (<span class="fu">exp</span>(x) <span class="op">-</span> <span class="fl">1</span> <span class="op">-</span> x) <span class="op">/</span> x<span class="op">^</span><span class="fl">2</span></span>
|
||
<span id="cb58-2"><a href="#cb58-2" aria-hidden="true" tabindex="-1"></a><span class="fu">limit</span>(<span class="fu">f</span>(x), x <span class="op">=></span> <span class="fl">0</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="39">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
\frac{1}{2}
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<p>Finally, for the third we define a new variable and proceed:</p>
|
||
<div class="cell" data-execution_count="40">
|
||
<div class="sourceCode cell-code" id="cb59"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb59-1"><a href="#cb59-1" aria-hidden="true" tabindex="-1"></a><span class="pp">@syms</span> rho<span class="op">::</span><span class="dt">real</span></span>
|
||
<span id="cb59-2"><a href="#cb59-2" aria-hidden="true" tabindex="-1"></a><span class="fu">limit</span>( (x<span class="op">^</span>(<span class="fl">1</span><span class="op">-</span>rho) <span class="op">-</span> <span class="fl">1</span>) <span class="op">/</span> (<span class="fl">1</span> <span class="op">-</span> rho), rho <span class="op">=></span> <span class="fl">1</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="40">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
\log{\left(x \right)}
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<p>This last limit demonstrates that the <code>limit</code> function of <code>SymPy</code> can readily evaluate limits that involve parameters, though at times some assumptions on the parameters may be needed, as was done through <code>rho::real</code></p>
|
||
<p>However, for some cases, the assumptions will not be enough, as they are broad. (E.g., something might be true for some values of the parameter and not others and these values aren’t captured in the assumptions.) So the user must be mindful that when parameters are involved, the answer may not reflect all possible cases.</p>
|
||
</section>
|
||
<section id="example-floating-point-conversion-issues" class="level5">
|
||
<h5 class="anchored" data-anchor-id="example-floating-point-conversion-issues">Example: floating point conversion issues</h5>
|
||
<p>The Gruntz <a href="http://www.cybertester.com/data/gruntz.pdf">algorithm</a> implemented in <code>SymPy</code> for symbolic limits is quite powerful. However, some care must be exercised to avoid undesirable conversions from exact values to floating point values.</p>
|
||
<p>In a previous example, we used <code>15//11</code> and not <code>15/11</code>, as the former converts to an <em>exact</em> symbolic value for use in <code>SymPy</code>, but the latter would be approximated in floating point <em>before</em> this conversion so the exactness would be lost.</p>
|
||
<p>To illustrate further, let’s look at the limit as <span class="math inline">\(x\)</span> goes to <span class="math inline">\(\pi/2\)</span> of <span class="math inline">\(j(x) = \cos(x) / (x - \pi/2)\)</span>. We follow our past practice:</p>
|
||
<div class="cell" data-execution_count="41">
|
||
<div class="sourceCode cell-code" id="cb60"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb60-1"><a href="#cb60-1" aria-hidden="true" tabindex="-1"></a><span class="fu">j</span>(x) <span class="op">=</span> <span class="fu">cos</span>(x) <span class="op">/</span> (x <span class="op">-</span> <span class="cn">pi</span><span class="op">/</span><span class="fl">2</span>)</span>
|
||
<span id="cb60-2"><a href="#cb60-2" aria-hidden="true" tabindex="-1"></a><span class="fu">j</span>(<span class="cn">pi</span><span class="op">/</span><span class="fl">2</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="41">
|
||
<pre><code>Inf</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>The value is not <code>NaN</code>, rather <code>Inf</code>. This is because <code>cos(pi/2)</code> is not exactly <span class="math inline">\(0\)</span> as it should be mathematically, as <code>pi/2</code> is rounded to a floating point number. This minor difference is important. If we try and correct for this by using <code>PI</code> we have:</p>
|
||
<div class="cell" data-execution_count="42">
|
||
<div class="sourceCode cell-code" id="cb62"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb62-1"><a href="#cb62-1" aria-hidden="true" tabindex="-1"></a><span class="fu">limit</span>(<span class="fu">j</span>(x), x <span class="op">=></span> PI<span class="op">/</span><span class="fl">2</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="42">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
0
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<p>The value is not right, as this simple graph suggests the limit is in fact <span class="math inline">\(-1\)</span>:</p>
|
||
<div class="cell" data-execution_count="43">
|
||
<div class="sourceCode cell-code" id="cb63"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb63-1"><a href="#cb63-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(j, <span class="cn">pi</span><span class="op">/</span><span class="fl">4</span>, <span class="fl">3</span>pi<span class="op">/</span><span class="fl">4</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="43">
|
||
<p><img src="limits_files/figure-html/cell-44-output-1.svg" class="img-fluid"></p>
|
||
</div>
|
||
</div>
|
||
<p>The difference between <code>pi</code> and <code>PI</code> can be significant, and though usually <code>pi</code> is silently converted to <code>PI</code>, it doesn’t happen here as the division by <code>2</code> happens first, which turns the symbol into an approximate floating point number. Hence, <code>SymPy</code> is giving the correct answer for the problem it is given, it just isn’t the problem we wanted to look at.</p>
|
||
<p>Trying again, being more aware of how <code>pi</code> and <code>PI</code> differ, we have:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="44">
|
||
<div class="sourceCode cell-code" id="cb64"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb64-1"><a href="#cb64-1" aria-hidden="true" tabindex="-1"></a><span class="fu">f</span>(x) <span class="op">=</span> <span class="fu">cos</span>(x) <span class="op">/</span> (x <span class="op">-</span> PI<span class="op">/</span><span class="fl">2</span>)</span>
|
||
<span id="cb64-2"><a href="#cb64-2" aria-hidden="true" tabindex="-1"></a><span class="fu">limit</span>(<span class="fu">f</span>(x), x <span class="op">=></span> PI<span class="op">/</span><span class="fl">2</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="44">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
-1
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<p>(The value <code>pi</code> is able to be exactly converted to <code>PI</code> when used in <code>SymPy</code>, as it is of type <code>Irrational</code>, and is not a floating point value. However, the expression <code>pi/2</code> converts <code>pi</code> to a floating point value and then divides by <code>2</code>, hence the loss of exactness when used symbolically.)</p>
|
||
</section>
|
||
<section id="example-left-and-right-limits" class="level5">
|
||
<h5 class="anchored" data-anchor-id="example-left-and-right-limits">Example: left and right limits</h5>
|
||
<p>Right and left limits will be discussed in the next section; here we give an example of the idea. The mathematical convention is to say a limit exists if both the left <em>and</em> right limits exist and are equal. Informally a right (left) limit at <span class="math inline">\(c\)</span> only considers values of <span class="math inline">\(x\)</span> less (more) than <span class="math inline">\(c\)</span>. The <code>limit</code> function of <code>SymPy</code> finds directional limits by default, a right limit, where <span class="math inline">\(x > c\)</span>.</p>
|
||
<p>The left limit can be found by passing the argument <code>dir="-"</code>. Passing <code>dir="+-"</code> (and not <code>"-+"</code>) will compute the mathematical limit, throwing an error in <code>Python</code> if no limit exists.</p>
|
||
<div class="cell" data-execution_count="45">
|
||
<div class="sourceCode cell-code" id="cb65"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb65-1"><a href="#cb65-1" aria-hidden="true" tabindex="-1"></a><span class="fu">limit</span>(<span class="fu">ceil</span>(x), x <span class="op">=></span> <span class="fl">0</span>), <span class="fu">limit</span>(<span class="fu">ceil</span>(x), x <span class="op">=></span> <span class="fl">0</span>, dir<span class="op">=</span><span class="st">"-"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="45">
|
||
<pre><code>(1, 0)</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>This accurately shows the limit does not exist mathematically, but <code>limit(ceil(x), x => 0)</code> does exist (as it finds a right limit).</p>
|
||
</section>
|
||
</section>
|
||
<section id="rules-for-limits" class="level2" data-number="18.4">
|
||
<h2 data-number="18.4" class="anchored" data-anchor-id="rules-for-limits"><span class="header-section-number">18.4</span> Rules for limits</h2>
|
||
<p>The <code>limit</code> function doesn’t compute limits from the definition, rather it applies some known facts about functions within a set of rules. Some of these rules are the following. Suppose the individual limits of <span class="math inline">\(f\)</span> and <span class="math inline">\(g\)</span> always exist (and are finite) below.</p>
|
||
<p><span class="math display">\[
|
||
\begin{align*}
|
||
\lim_{x \rightarrow c} (a \cdot f(x) + b \cdot g(x)) &= a \cdot
|
||
\lim_{x \rightarrow c} f(x) + b \cdot \lim_{x \rightarrow c} g(x)
|
||
&\\
|
||
%%
|
||
\lim_{x \rightarrow c} f(x) \cdot g(x) &= \lim_{x \rightarrow c}
|
||
f(x) \cdot \lim_{x \rightarrow c} g(x)
|
||
&\\
|
||
%%
|
||
\lim_{x \rightarrow c} \frac{f(x)}{g(x)} &=
|
||
\frac{\lim_{x \rightarrow c} f(x)}{\lim_{x \rightarrow c} g(x)}
|
||
&(\text{provided }\lim_{x \rightarrow c} g(x) \neq 0)\\
|
||
\end{align*}
|
||
\]</span></p>
|
||
<p>These are verbally described as follows, when the individual limits exist and are finite then:</p>
|
||
<ul>
|
||
<li>Limits involving sums, differences or scalar multiples of functions <em>exist</em> <strong>and</strong> can be <strong>computed</strong> by first doing the individual limits and then combining the answers appropriately.</li>
|
||
<li>Limits of products exist and can be found by computing the limits of the individual factors and then combining.</li>
|
||
<li>Limits of ratios <em>exist</em> and can be found by computing the limit of the individual terms and then dividing <strong>provided</strong> you don’t divide by <span class="math inline">\(0\)</span>. The last part is really important, as this rule is no help with the common indeterminate form <span class="math inline">\(0/0\)</span></li>
|
||
</ul>
|
||
<p>In addition, consider the composition:</p>
|
||
<p><span class="math display">\[
|
||
\lim_{x \rightarrow c} f(g(x))
|
||
\]</span></p>
|
||
<p>Suppose that</p>
|
||
<ul>
|
||
<li>The outer limit, <span class="math inline">\(\lim_{x \rightarrow b} f(x) = L\)</span>, exists, and</li>
|
||
<li>the inner limit, <span class="math inline">\(\lim_{x \rightarrow c} g(x) = b\)</span>, exists <strong>and</strong></li>
|
||
<li>for some neighborhood around <span class="math inline">\(c\)</span> (not including <span class="math inline">\(c\)</span>) <span class="math inline">\(g(x)\)</span> is not <span class="math inline">\(b\)</span>,</li>
|
||
</ul>
|
||
<p>Then the limit exists and equals <span class="math inline">\(L\)</span>:</p>
|
||
<p><span class="math inline">\(\lim_{x \rightarrow c} f(g(x)) = \lim_{u \rightarrow b} f(u) = L.\)</span></p>
|
||
<p>An alternative, is to assume <span class="math inline">\(f(x)\)</span> is defined at <span class="math inline">\(b\)</span> and equal to <span class="math inline">\(L\)</span> (which is the definition of continuity), but that isn’t the assumption above, hence the need to exclude <span class="math inline">\(g\)</span> from taking on a value of <span class="math inline">\(b\)</span> (where <span class="math inline">\(f\)</span> may not be defined) near <span class="math inline">\(c\)</span>.</p>
|
||
<p>These rules, together with the fact that our basic algebraic functions have limits that can be found by simple evaluation, mean that many limits are easy to compute.</p>
|
||
<section id="example-composition" class="level5">
|
||
<h5 class="anchored" data-anchor-id="example-composition">Example: composition</h5>
|
||
<p>For example, consider for some non-zero <span class="math inline">\(k\)</span> the following limit:</p>
|
||
<p><span class="math display">\[
|
||
\lim_{x \rightarrow 0} \frac{\sin(kx)}{x}.
|
||
\]</span></p>
|
||
<p>This is clearly related to the function <span class="math inline">\(f(x) = \sin(x)/x\)</span>, which has a limit of <span class="math inline">\(1\)</span> as <span class="math inline">\(x \rightarrow 0\)</span>. We see <span class="math inline">\(g(x) = k f(kx)\)</span> is the limit in question. As <span class="math inline">\(kx \rightarrow 0\)</span>, though not taking a value of <span class="math inline">\(0\)</span> except when <span class="math inline">\(x=0\)</span>, the limit above is <span class="math inline">\(k \lim_{x \rightarrow 0} f(kx) = k \lim_{u \rightarrow 0} f(u) = 1\)</span>.</p>
|
||
<p>Basically when taking a limit as <span class="math inline">\(x\)</span> goes to <span class="math inline">\(0\)</span> we can multiply <span class="math inline">\(x\)</span> by any constant and figure out the limit for that. (It is as though we “go to” <span class="math inline">\(0\)</span> faster or slower. but are still going to <span class="math inline">\(0\)</span>.</p>
|
||
<p>Similarly,</p>
|
||
<p><span class="math display">\[
|
||
\lim_{x \rightarrow 0} \frac{\sin(x^2)}{x^2} = 1,
|
||
\]</span></p>
|
||
<p>as this is the limit of <span class="math inline">\(f(g(x))\)</span> with <span class="math inline">\(f\)</span> as above and <span class="math inline">\(g(x) = x^2\)</span>. We need <span class="math inline">\(x \rightarrow 0\)</span>, <span class="math inline">\(g\)</span>is only <span class="math inline">\(0\)</span> at <span class="math inline">\(x=0\)</span>, which is the case.</p>
|
||
</section>
|
||
<section id="example-products" class="level5">
|
||
<h5 class="anchored" data-anchor-id="example-products">Example: products</h5>
|
||
<p>Consider this complicated limit found on this <a href="http://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule">Wikipedia</a> page.</p>
|
||
<p><span class="math display">\[
|
||
\lim_{x \rightarrow 1/2} \frac{\sin(\pi x)}{\pi x} \cdot \frac{\cos(\pi x)}{1 - (2x)^2}.
|
||
\]</span></p>
|
||
<p>We know the first factor has a limit found by evaluation: <span class="math inline">\(2/\pi\)</span>, so it is really just a constant. The second we can compute:</p>
|
||
<div class="cell" data-execution_count="46">
|
||
<div class="sourceCode cell-code" id="cb67"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb67-1"><a href="#cb67-1" aria-hidden="true" tabindex="-1"></a><span class="fu">l</span>(x) <span class="op">=</span> <span class="fu">cos</span>(PI<span class="op">*</span>x) <span class="op">/</span> (<span class="fl">1</span> <span class="op">-</span> (<span class="fl">2</span>x)<span class="op">^</span><span class="fl">2</span>)</span>
|
||
<span id="cb67-2"><a href="#cb67-2" aria-hidden="true" tabindex="-1"></a><span class="fu">limit</span>(l, <span class="fl">1</span><span class="op">//</span><span class="fl">2</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="46">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
\frac{\pi}{4}
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<p>Putting together, we would get <span class="math inline">\(1/2\)</span>. Which we could have done directly in this case:</p>
|
||
<div class="cell" data-execution_count="47">
|
||
<div class="sourceCode cell-code" id="cb68"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb68-1"><a href="#cb68-1" aria-hidden="true" tabindex="-1"></a><span class="fu">limit</span>(<span class="fu">sin</span>(PI<span class="op">*</span>x)<span class="op">/</span>(PI<span class="op">*</span>x) <span class="op">*</span> <span class="fu">l</span>(x), x <span class="op">=></span> <span class="fl">1</span><span class="op">//</span><span class="fl">2</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="47">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
\frac{1}{2}
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="example-ratios" class="level5">
|
||
<h5 class="anchored" data-anchor-id="example-ratios">Example: ratios</h5>
|
||
<p>Consider again the limit of <span class="math inline">\(\cos(\pi x) / (1 - (2x)^2)\)</span> at <span class="math inline">\(c=1/2\)</span>. A graph of both the top and bottom functions shows the indeterminate, <span class="math inline">\(0/0\)</span>, form:</p>
|
||
<div class="cell" data-execution_count="48">
|
||
<div class="sourceCode cell-code" id="cb69"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb69-1"><a href="#cb69-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(<span class="fu">cos</span>(<span class="cn">pi</span><span class="op">*</span>x), <span class="fl">0.4</span>, <span class="fl">0.6</span>)</span>
|
||
<span id="cb69-2"><a href="#cb69-2" aria-hidden="true" tabindex="-1"></a><span class="fu">plot!</span>(<span class="fl">1</span> <span class="op">-</span> (<span class="fl">2</span>x)<span class="op">^</span><span class="fl">2</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="48">
|
||
<p><img src="limits_files/figure-html/cell-49-output-1.svg" class="img-fluid"></p>
|
||
</div>
|
||
</div>
|
||
<p>However, following Euler’s insight that <span class="math inline">\(\sin(x)/x\)</span> will have a limit at <span class="math inline">\(0\)</span> of <span class="math inline">\(1\)</span> as <span class="math inline">\(\sin(x) \approx x\)</span>, and <span class="math inline">\(x/x\)</span> has a limit of <span class="math inline">\(1\)</span> at <span class="math inline">\(c=0\)</span>, we can see that <span class="math inline">\(\cos(\pi x)\)</span> looks like <span class="math inline">\(-\pi\cdot (x - 1/2)\)</span> and <span class="math inline">\((1 - (2x)^2)\)</span> looks like <span class="math inline">\(-4(x-1/2)\)</span> around <span class="math inline">\(x=1/2\)</span>:</p>
|
||
<div class="cell" data-execution_count="49">
|
||
<div class="sourceCode cell-code" id="cb70"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb70-1"><a href="#cb70-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(<span class="fu">cos</span>(<span class="cn">pi</span><span class="op">*</span>x), <span class="fl">0.4</span>, <span class="fl">0.6</span>)</span>
|
||
<span id="cb70-2"><a href="#cb70-2" aria-hidden="true" tabindex="-1"></a><span class="fu">plot!</span>(<span class="fu">-pi*</span>(x <span class="op">-</span> <span class="fl">1</span><span class="op">/</span><span class="fl">2</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="49">
|
||
<p><img src="limits_files/figure-html/cell-50-output-1.svg" class="img-fluid"></p>
|
||
</div>
|
||
</div>
|
||
<div class="cell" data-execution_count="50">
|
||
<div class="sourceCode cell-code" id="cb71"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb71-1"><a href="#cb71-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(<span class="fl">1</span> <span class="op">-</span> (<span class="fl">2</span>x)<span class="op">^</span><span class="fl">2</span>, <span class="fl">0.4</span>, <span class="fl">0.6</span>)</span>
|
||
<span id="cb71-2"><a href="#cb71-2" aria-hidden="true" tabindex="-1"></a><span class="fu">plot!</span>(<span class="fu">-4</span>(x <span class="op">-</span> <span class="fl">1</span><span class="op">/</span><span class="fl">2</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="50">
|
||
<p><img src="limits_files/figure-html/cell-51-output-1.svg" class="img-fluid"></p>
|
||
</div>
|
||
</div>
|
||
<p>So around <span class="math inline">\(c=1/2\)</span> the ratio should look like <span class="math inline">\(-\pi (x-1/2) / ( -4(x - 1/2)) = \pi/4\)</span>, which indeed it does, as that is the limit.</p>
|
||
<p>This is the basis of L’Hôpital’s rule, which we will return to once the derivative is discussed.</p>
|
||
</section>
|
||
<section id="example-sums" class="level5">
|
||
<h5 class="anchored" data-anchor-id="example-sums">Example: sums</h5>
|
||
<p>If it is known that the following limit exists by some means:</p>
|
||
<p><span class="math display">\[
|
||
L = 0 = \lim_{x \rightarrow 0} \frac{e^{\csc(x)}}{e^{\cot(x)}} - (1 + \frac{1}{2}x + \frac{1}{8}x^2)
|
||
\]</span></p>
|
||
<p>Then this limit will exist</p>
|
||
<p><span class="math display">\[
|
||
M = \lim_{x \rightarrow 0} \frac{e^{\csc(x)}}{e^{\cot(x)}}
|
||
\]</span></p>
|
||
<p>Why? We can express the function <span class="math inline">\(e^{\csc(x)}/e^{\cot(x)}\)</span> as the above function plus the polynomial <span class="math inline">\(1 + x/2 + x^2/8\)</span>. The above is then the sum of two functions whose limits exist and are finite, hence, we can conclude that <span class="math inline">\(M = 0 + 1\)</span>.</p>
|
||
</section>
|
||
<section id="the-squeeze-theorem" class="level3" data-number="18.4.1">
|
||
<h3 data-number="18.4.1" class="anchored" data-anchor-id="the-squeeze-theorem"><span class="header-section-number">18.4.1</span> The <a href="http://en.wikipedia.org/wiki/Squeeze_theorem">squeeze</a> theorem</h3>
|
||
<p>We note one more limit law. Suppose we wish to compute <span class="math inline">\(\lim_{x \rightarrow c}f(x)\)</span> and we have two other functions, <span class="math inline">\(l\)</span> and <span class="math inline">\(u\)</span>, satisfying:</p>
|
||
<ul>
|
||
<li>for all <span class="math inline">\(x\)</span> near <span class="math inline">\(c\)</span> (possibly not including <span class="math inline">\(c\)</span>) <span class="math inline">\(l(x) \leq f(x) \leq u(x)\)</span>.</li>
|
||
<li>These limits exist and are equal: <span class="math inline">\(L = \lim_{x \rightarrow c} l(x) = \lim_{x \rightarrow c} u(x)\)</span>.</li>
|
||
</ul>
|
||
<p>Then the limit of <span class="math inline">\(f\)</span> must also be <span class="math inline">\(L\)</span>.</p>
|
||
<div class="cell" data-hold="true" data-execution_count="51">
|
||
<div class="cell-output cell-output-display" data-execution_count="51">
|
||
<div class="d-flex justify-content-center"> <figure class="figure"> <img 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" class="card-img-top figure-img" alt="A Figure">
|
||
<figcaption class="figure-caption"><div class="markdown"><p>As \(x\) goes to \(0\), the values of \(sin(x)/x\) are squeezed between \(\cos(x)\) and \(1\) which both converge to \(1\).</p>
|
||
</div> </figcaption>
|
||
</figure>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
</section>
|
||
<section id="limits-from-the-definition" class="level2" data-number="18.5">
|
||
<h2 data-number="18.5" class="anchored" data-anchor-id="limits-from-the-definition"><span class="header-section-number">18.5</span> Limits from the definition</h2>
|
||
<p>The formal definition of a limit involves clarifying what it means for <span class="math inline">\(f(x)\)</span> to be “close to <span class="math inline">\(L\)</span>” when <span class="math inline">\(x\)</span> is “close to <span class="math inline">\(c\)</span>”. These are quantified by the inequalities <span class="math inline">\(0 < \lvert x-c\rvert < \delta\)</span> and the <span class="math inline">\(\lvert f(x) - L\rvert < \epsilon\)</span>. The second does not have the restriction that it is greater than <span class="math inline">\(0\)</span>, as indeed <span class="math inline">\(f(x)\)</span> can equal <span class="math inline">\(L\)</span>. The order is important: it says for any idea of close for <span class="math inline">\(f(x)\)</span> to <span class="math inline">\(L\)</span>, an idea of close must be found for <span class="math inline">\(x\)</span> to <span class="math inline">\(c\)</span>.</p>
|
||
<p>The key is identifying a value for <span class="math inline">\(\delta\)</span> for a given value of <span class="math inline">\(\epsilon\)</span>.</p>
|
||
<p>A simple case is the linear case. Consider the function <span class="math inline">\(f(x) = 3x + 2\)</span>. Verify that the limit at <span class="math inline">\(c=1\)</span> is <span class="math inline">\(5\)</span>.</p>
|
||
<p>We show “numerically” that <span class="math inline">\(\delta = \epsilon/3\)</span>.</p>
|
||
<div class="cell" data-hold="true" data-execution_count="52">
|
||
<div class="sourceCode cell-code" id="cb72"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb72-1"><a href="#cb72-1" aria-hidden="true" tabindex="-1"></a><span class="fu">f</span>(x) <span class="op">=</span> <span class="fl">3</span>x <span class="op">+</span> <span class="fl">2</span></span>
|
||
<span id="cb72-2"><a href="#cb72-2" aria-hidden="true" tabindex="-1"></a>c, L <span class="op">=</span> <span class="fl">1</span>, <span class="fl">5</span></span>
|
||
<span id="cb72-3"><a href="#cb72-3" aria-hidden="true" tabindex="-1"></a>epsilon <span class="op">=</span> <span class="fu">rand</span>() <span class="co"># some number in (0,1)</span></span>
|
||
<span id="cb72-4"><a href="#cb72-4" aria-hidden="true" tabindex="-1"></a>delta <span class="op">=</span> epsilon <span class="op">/</span> <span class="fl">3</span></span>
|
||
<span id="cb72-5"><a href="#cb72-5" aria-hidden="true" tabindex="-1"></a>xs <span class="op">=</span> c <span class="op">.+</span> delta <span class="op">*</span> <span class="fu">rand</span>(<span class="fl">100</span>) <span class="co"># 100 numbers, c < x < c + delta</span></span>
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<span id="cb72-6"><a href="#cb72-6" aria-hidden="true" tabindex="-1"></a>as <span class="op">=</span> [<span class="fu">abs</span>(<span class="fu">f</span>(x) <span class="op">-</span> L) <span class="op"><</span> epsilon for x <span class="kw">in</span> xs]</span>
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<span id="cb72-7"><a href="#cb72-7" aria-hidden="true" tabindex="-1"></a><span class="fu">all</span>(as) <span class="co"># are all the as true?</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<div class="cell-output cell-output-display" data-execution_count="52">
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<pre><code>true</code></pre>
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</div>
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</div>
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<p>These lines produce a random <span class="math inline">\(\epsilon\)</span>, the resulting <span class="math inline">\(\delta\)</span>, and then verify for 100 numbers within <span class="math inline">\((c, c+\delta)\)</span> that the inequality <span class="math inline">\(\lvert f(x) - L \rvert < \epsilon\)</span> holds for each. Running them again and again should always produce <code>true</code> if <span class="math inline">\(L\)</span> is the limit and <span class="math inline">\(\delta\)</span> is chosen properly.</p>
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<p>(Of course, we should also verify values to the left of <span class="math inline">\(c\)</span>.)</p>
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||
<p>(The random numbers are technically in <span class="math inline">\([0,1)\)</span>, so in theory <code>epsilon</code> could be <code>0</code>. So the above approach would be more solid if some guard, such as <code>epsilon = max(eps(), rand())</code>, was used. As the formal definition is the domain of paper-and-pencil, we don’t fuss.)</p>
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<p>In this case, <span class="math inline">\(\delta\)</span> is easy to guess, as the function is linear and has slope <span class="math inline">\(3\)</span>. This basically says the <span class="math inline">\(y\)</span> scale is 3 times the <span class="math inline">\(x\)</span> scale. For non-linear functions, finding <span class="math inline">\(\delta\)</span> for a given <span class="math inline">\(\epsilon\)</span> can be a challenge. For the function <span class="math inline">\(f(x) = x^3\)</span>, illustrated below, a value of <span class="math inline">\(\delta=\epsilon^{1/3}\)</span> is used for <span class="math inline">\(c=0\)</span>:</p>
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<div class="cell" data-cache="true" data-hold="true" data-execution_count="53">
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<div class="cell-output cell-output-display" data-execution_count="53">
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<div class="d-flex justify-content-center"> <figure class="figure"> <img 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" class="card-img-top figure-img" alt="A Figure">
|
||
<figcaption class="figure-caption"><div class="markdown"><p>Demonstration of \(\epsilon\)-\(\delta\) proof of \(\lim_{x \rightarrow 0} x^3 = 0\). For any \(\epsilon>0\) (the orange lines) there exists a \(\delta>0\) (the red lines of the box) for which the function \(f(x)\) does not leave the top or bottom of the box (except possibly at the edges). In this example \(\delta^3=\epsilon\).</p>
|
||
</div> </figcaption>
|
||
</figure>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="questions" class="level2" data-number="18.6">
|
||
<h2 data-number="18.6" class="anchored" data-anchor-id="questions"><span class="header-section-number">18.6</span> Questions</h2>
|
||
<section id="question" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question">Question</h6>
|
||
<p>From the graph, find the limit:</p>
|
||
<p><span class="math display">\[
|
||
L = \lim_{x\rightarrow 1} \frac{x^2−3x+2}{x^2−6x+5}
|
||
\]</span></p>
|
||
<div class="cell" data-hold="true" data-execution_count="54">
|
||
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|
||
<p><img src="limits_files/figure-html/cell-55-output-1.svg" class="img-fluid"></p>
|
||
</div>
|
||
</div>
|
||
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|
||
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|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_133682357453776880")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-1" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-1">Question</h6>
|
||
<p>From the graph, find the limit <span class="math inline">\(L\)</span>:</p>
|
||
<p><span class="math display">\[
|
||
L = \lim_{x \rightarrow -2} \frac{x}{x+1} \frac{x^2}{x^2 + 4}
|
||
\]</span></p>
|
||
<div class="cell" data-hold="true" data-execution_count="56">
|
||
<div class="cell-output cell-output-display" data-execution_count="56">
|
||
<p><img src="limits_files/figure-html/cell-57-output-1.svg" class="img-fluid"></p>
|
||
</div>
|
||
</div>
|
||
<div class="cell" data-hold="true" data-execution_count="57">
|
||
<div class="cell-output cell-output-display" data-execution_count="57">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="11772326381591189476" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_11772326381591189476">
|
||
<div style="padding-top: 5px">
|
||
<br>
|
||
<div class="input-group">
|
||
<input id="11772326381591189476" type="number" class="form-control" placeholder="Numeric answer">
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="11772326381591189476_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.getElementById("11772326381591189476").addEventListener("change", function() {
|
||
var correct = (Math.abs(this.value - 1.0) <= 0.1);
|
||
var msgBox = document.getElementById('11772326381591189476_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_11772326381591189476")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_11772326381591189476")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-2" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-2">Question</h6>
|
||
<p>Graphically investigate the limit</p>
|
||
<p><span class="math display">\[
|
||
L = \lim_{x \rightarrow 0} \frac{e^x - 1}{x}.
|
||
\]</span></p>
|
||
<p>What is the value of <span class="math inline">\(L\)</span>?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="58">
|
||
<div class="cell-output cell-output-display" data-execution_count="58">
|
||
<p><img src="limits_files/figure-html/cell-59-output-1.svg" class="img-fluid"></p>
|
||
</div>
|
||
</div>
|
||
<div class="cell" data-hold="true" data-execution_count="59">
|
||
<div class="cell-output cell-output-display" data-execution_count="59">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="10449633555273883682" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_10449633555273883682">
|
||
<div style="padding-top: 5px">
|
||
<br>
|
||
<div class="input-group">
|
||
<input id="10449633555273883682" type="number" class="form-control" placeholder="Numeric answer">
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="10449633555273883682_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.getElementById("10449633555273883682").addEventListener("change", function() {
|
||
var correct = (Math.abs(this.value - 1) <= 0.1);
|
||
var msgBox = document.getElementById('10449633555273883682_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_10449633555273883682")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_10449633555273883682")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-3" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-3">Question</h6>
|
||
<p>Graphically investigate the limit</p>
|
||
<p><span class="math display">\[
|
||
\lim_{x \rightarrow 0} \frac{\cos(x) - 1}{x}.
|
||
\]</span></p>
|
||
<p>The limit exists, what is the value?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="60">
|
||
<div class="cell-output cell-output-display" data-execution_count="60">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="16139820609448316164" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_16139820609448316164">
|
||
<div style="padding-top: 5px">
|
||
<br>
|
||
<div class="input-group">
|
||
<input id="16139820609448316164" type="number" class="form-control" placeholder="Numeric answer">
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="16139820609448316164_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.getElementById("16139820609448316164").addEventListener("change", function() {
|
||
var correct = (Math.abs(this.value - 0) <= 0.01);
|
||
var msgBox = document.getElementById('16139820609448316164_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_16139820609448316164")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_16139820609448316164")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-4" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-4">Question</h6>
|
||
<p>Select the graph for which there is no limit at <span class="math inline">\(a\)</span>.</p>
|
||
<div class="cell" data-hold="true" data-execution_count="61">
|
||
<div class="cell-output cell-output-display" data-execution_count="61">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="9715153777918695850" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_9715153777918695850">
|
||
<div style="padding-top: 5px">
|
||
<div>
|
||
<img id="hotspot_9715153777918695850" src="data:image/gif;base64,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|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="9715153777918695850_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.getElementById("hotspot_9715153777918695850").addEventListener("click", function(e) {
|
||
var u = e.offsetX;
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||
var v = e.offsetY;
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||
var w = this.offsetWidth;
|
||
var h = this.offsetHeight
|
||
|
||
var x = u/w;
|
||
var y = (h-v)/h
|
||
|
||
var correct = (x >= 0.5 && x <= 1.0 && y >= 0.5 && y <= 1.0);
|
||
var msgBox = document.getElementById('9715153777918695850_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_9715153777918695850")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_9715153777918695850")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
|
||
})
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-5" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-5">Question</h6>
|
||
<p>The following limit is commonly used:</p>
|
||
<p><span class="math display">\[
|
||
\lim_{h \rightarrow 0} \frac{e^{x + h} - e^x}{h} = L.
|
||
\]</span></p>
|
||
<p>Factoring out <span class="math inline">\(e^x\)</span> from the top and using rules of limits this becomes,</p>
|
||
<p><span class="math display">\[
|
||
L = e^x \lim_{h \rightarrow 0} \frac{e^h - 1}{h}.
|
||
\]</span></p>
|
||
<p>What is <span class="math inline">\(L\)</span>?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="62">
|
||
<div class="cell-output cell-output-display" data-execution_count="62">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="13004426570995400530" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_13004426570995400530">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_13004426570995400530_1">
|
||
<input class="form-check-input" type="radio" name="radio_13004426570995400530" id="radio_13004426570995400530_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
\(0\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_13004426570995400530_2">
|
||
<input class="form-check-input" type="radio" name="radio_13004426570995400530" id="radio_13004426570995400530_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
\(1\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_13004426570995400530_3">
|
||
<input class="form-check-input" type="radio" name="radio_13004426570995400530" id="radio_13004426570995400530_3" value="3">
|
||
|
||
<span class="label-body px-1">
|
||
\(e^x\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="13004426570995400530_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_13004426570995400530"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 3;
|
||
var msgBox = document.getElementById('13004426570995400530_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_13004426570995400530")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_13004426570995400530")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-6" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-6">Question</h6>
|
||
<p>The following limit is commonly used:</p>
|
||
<p><span class="math display">\[
|
||
\lim_{h \rightarrow 0} \frac{\sin(x + h) - \sin(x)}{h} = L.
|
||
\]</span></p>
|
||
<p>The answer should depend on <span class="math inline">\(x\)</span>, though it is possible it is a constant. Using a double angle formula and the rules of limits, this can be written as:</p>
|
||
<p><span class="math display">\[
|
||
L = \cos(x) \lim_{h \rightarrow 0}\frac{\sin(h)}{h} + \sin(x) \lim_{h \rightarrow 0}\frac{\cos(h)-1}{h}.
|
||
\]</span></p>
|
||
<p>Using the last result, what is the value of <span class="math inline">\(L\)</span>?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="63">
|
||
<div class="cell-output cell-output-display" data-execution_count="63">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="12989662549651803021" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_12989662549651803021">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_12989662549651803021_1">
|
||
<input class="form-check-input" type="radio" name="radio_12989662549651803021" id="radio_12989662549651803021_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
\(\sin(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_12989662549651803021_2">
|
||
<input class="form-check-input" type="radio" name="radio_12989662549651803021" id="radio_12989662549651803021_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
\(\cos(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_12989662549651803021_3">
|
||
<input class="form-check-input" type="radio" name="radio_12989662549651803021" id="radio_12989662549651803021_3" value="3">
|
||
|
||
<span class="label-body px-1">
|
||
\(\sin(h)/h\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_12989662549651803021_4">
|
||
<input class="form-check-input" type="radio" name="radio_12989662549651803021" id="radio_12989662549651803021_4" value="4">
|
||
|
||
<span class="label-body px-1">
|
||
\(0\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_12989662549651803021_5">
|
||
<input class="form-check-input" type="radio" name="radio_12989662549651803021" id="radio_12989662549651803021_5" value="5">
|
||
|
||
<span class="label-body px-1">
|
||
\(1\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="12989662549651803021_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_12989662549651803021"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 2;
|
||
var msgBox = document.getElementById('12989662549651803021_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_12989662549651803021")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_12989662549651803021")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-7" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-7">Question</h6>
|
||
<p>Find the limit as <span class="math inline">\(x\)</span> goes to <span class="math inline">\(2\)</span> of</p>
|
||
<p><span class="math display">\[
|
||
f(x) = \frac{3x^2 - x -10}{x^2 - 4}
|
||
\]</span></p>
|
||
<div class="cell" data-hold="true" data-execution_count="64">
|
||
<div class="cell-output cell-output-display" data-execution_count="64">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="12345166601958729083" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_12345166601958729083">
|
||
<div style="padding-top: 5px">
|
||
<br>
|
||
<div class="input-group">
|
||
<input id="12345166601958729083" type="number" class="form-control" placeholder="Numeric answer">
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="12345166601958729083_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.getElementById("12345166601958729083").addEventListener("change", function() {
|
||
var correct = (Math.abs(this.value - 2.75) <= 0.001);
|
||
var msgBox = document.getElementById('12345166601958729083_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_12345166601958729083")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_12345166601958729083")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-8" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-8">Question</h6>
|
||
<p>Find the limit as <span class="math inline">\(x\)</span> goes to <span class="math inline">\(-2\)</span> of</p>
|
||
<p><span class="math display">\[
|
||
f(x) = \frac{\frac{1}{x} + \frac{1}{2}}{x^3 + 8}
|
||
\]</span></p>
|
||
<div class="cell" data-hold="true" data-execution_count="65">
|
||
<div class="cell-output cell-output-display" data-execution_count="65">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="13749159775220317193" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_13749159775220317193">
|
||
<div style="padding-top: 5px">
|
||
<br>
|
||
<div class="input-group">
|
||
<input id="13749159775220317193" type="number" class="form-control" placeholder="Numeric answer">
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="13749159775220317193_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.getElementById("13749159775220317193").addEventListener("change", function() {
|
||
var correct = (Math.abs(this.value - -0.020833333333333332) <= 0.001);
|
||
var msgBox = document.getElementById('13749159775220317193_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_13749159775220317193")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_13749159775220317193")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-9" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-9">Question</h6>
|
||
<p>Find the limit as <span class="math inline">\(x\)</span> goes to <span class="math inline">\(27\)</span> of</p>
|
||
<p><span class="math display">\[
|
||
f(x) = \frac{x - 27}{x^{1/3} - 3}
|
||
\]</span></p>
|
||
<div class="cell" data-hold="true" data-execution_count="66">
|
||
<div class="cell-output cell-output-display" data-execution_count="66">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="1698731682176548354" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_1698731682176548354">
|
||
<div style="padding-top: 5px">
|
||
<br>
|
||
<div class="input-group">
|
||
<input id="1698731682176548354" type="number" class="form-control" placeholder="Numeric answer">
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="1698731682176548354_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.getElementById("1698731682176548354").addEventListener("change", function() {
|
||
var correct = (Math.abs(this.value - 27) <= 0);
|
||
var msgBox = document.getElementById('1698731682176548354_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_1698731682176548354")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_1698731682176548354")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-10" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-10">Question</h6>
|
||
<p>Find the limit</p>
|
||
<p><span class="math display">\[
|
||
L = \lim_{x \rightarrow \pi/2} \frac{\tan (2x)}{x - \pi/2}
|
||
\]</span></p>
|
||
<div class="cell" data-hold="true" data-execution_count="67">
|
||
<div class="cell-output cell-output-display" data-execution_count="67">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="16259297656105099060" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_16259297656105099060">
|
||
<div style="padding-top: 5px">
|
||
<br>
|
||
<div class="input-group">
|
||
<input id="16259297656105099060" type="number" class="form-control" placeholder="Numeric answer">
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="16259297656105099060_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.getElementById("16259297656105099060").addEventListener("change", function() {
|
||
var correct = (Math.abs(this.value - 2) <= 0);
|
||
var msgBox = document.getElementById('16259297656105099060_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_16259297656105099060")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_16259297656105099060")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-11" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-11">Question</h6>
|
||
<p>The limit of <span class="math inline">\(\sin(x)/x\)</span> at <span class="math inline">\(0\)</span> has a numeric value. This depends upon the fact that <span class="math inline">\(x\)</span> is measured in radians. Try to find this limit: <code>limit(sind(x)/x, x => 0)</code>. What is the value?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="68">
|
||
<div class="cell-output cell-output-display" data-execution_count="68">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="1934252921752046666" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_1934252921752046666">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_1934252921752046666_1">
|
||
<input class="form-check-input" type="radio" name="radio_1934252921752046666" id="radio_1934252921752046666_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
<code>0</code>
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_1934252921752046666_2">
|
||
<input class="form-check-input" type="radio" name="radio_1934252921752046666" id="radio_1934252921752046666_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
<code>1</code>
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_1934252921752046666_3">
|
||
<input class="form-check-input" type="radio" name="radio_1934252921752046666" id="radio_1934252921752046666_3" value="3">
|
||
|
||
<span class="label-body px-1">
|
||
<code>180/pi</code>
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_1934252921752046666_4">
|
||
<input class="form-check-input" type="radio" name="radio_1934252921752046666" id="radio_1934252921752046666_4" value="4">
|
||
|
||
<span class="label-body px-1">
|
||
<code>pi/180</code>
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="1934252921752046666_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_1934252921752046666"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 4;
|
||
var msgBox = document.getElementById('1934252921752046666_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_1934252921752046666")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_1934252921752046666")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
<p>What is the limit <code>limit(sinpi(x)/x, x => 0)</code>?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="69">
|
||
<div class="cell-output cell-output-display" data-execution_count="69">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="6236489184751078397" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_6236489184751078397">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_6236489184751078397_1">
|
||
<input class="form-check-input" type="radio" name="radio_6236489184751078397" id="radio_6236489184751078397_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
<code>1</code>
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_6236489184751078397_2">
|
||
<input class="form-check-input" type="radio" name="radio_6236489184751078397" id="radio_6236489184751078397_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
<code>1/pi</code>
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_6236489184751078397_3">
|
||
<input class="form-check-input" type="radio" name="radio_6236489184751078397" id="radio_6236489184751078397_3" value="3">
|
||
|
||
<span class="label-body px-1">
|
||
<code>0</code>
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_6236489184751078397_4">
|
||
<input class="form-check-input" type="radio" name="radio_6236489184751078397" id="radio_6236489184751078397_4" value="4">
|
||
|
||
<span class="label-body px-1">
|
||
<code>pi</code>
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="6236489184751078397_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_6236489184751078397"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 4;
|
||
var msgBox = document.getElementById('6236489184751078397_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_6236489184751078397")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_6236489184751078397")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-limit-properties" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-limit-properties">Question: limit properties</h6>
|
||
<p>There are several properties of limits that allow one to break down more complicated problems into smaller subproblems. For example,</p>
|
||
<p><span class="math display">\[
|
||
\lim (f(x) + g(x)) = \lim f(x) + \lim g(x)
|
||
\]</span></p>
|
||
<p>is notation to indicate that one can take a limit of the sum of two function or take the limit of each first, then add and the answer will be unchanged, provided all the limits in question exist.</p>
|
||
<p>Use one or the either to find the limit of <span class="math inline">\(f(x) = \sin(x) + \tan(x) + \cos(x)\)</span> as <span class="math inline">\(x\)</span> goes to <span class="math inline">\(0\)</span>.</p>
|
||
<div class="cell" data-hold="true" data-execution_count="70">
|
||
<div class="cell-output cell-output-display" data-execution_count="70">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="11614836563914983046" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_11614836563914983046">
|
||
<div style="padding-top: 5px">
|
||
<br>
|
||
<div class="input-group">
|
||
<input id="11614836563914983046" type="number" class="form-control" placeholder="Numeric answer">
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="11614836563914983046_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.getElementById("11614836563914983046").addEventListener("change", function() {
|
||
var correct = (Math.abs(this.value - 1.0) <= 1.0e-5);
|
||
var msgBox = document.getElementById('11614836563914983046_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_11614836563914983046")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_11614836563914983046")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-12" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-12">Question</h6>
|
||
<p>The key assumption made above in being able to write</p>
|
||
<p><span class="math display">\[
|
||
\lim_{x\rightarrow c} f(g(x)) = L,
|
||
\]</span></p>
|
||
<p>when <span class="math inline">\(\lim_{x\rightarrow b} f(x) = L\)</span> and <span class="math inline">\(\lim_{x\rightarrow c}g(x) = b\)</span> is <em>continuity</em>.</p>
|
||
<p>This <a href="https://en.wikipedia.org/wiki/Limit_of_a_function#Limits_of_compositions_of_functions">example</a> shows why it is important.</p>
|
||
<p>Take</p>
|
||
<p><span class="math display">\[
|
||
f(x) = \begin{cases}
|
||
0 & x \neq 0\\
|
||
1 & x = 0
|
||
\end{cases}
|
||
\]</span></p>
|
||
<p>We have <span class="math inline">\(\lim_{x\rightarrow 0}f(x) = 0\)</span>, as <span class="math inline">\(0\)</span> is clearly a removable discontinuity. So were the above applicable we would have <span class="math inline">\(\lim_{x \rightarrow 0}f(f(x)) = 0\)</span>. But this is not true. What is the limit at <span class="math inline">\(0\)</span> of <span class="math inline">\(f(f(x))\)</span>?</p>
|
||
<div class="cell" data-execution_count="71">
|
||
<div class="sourceCode cell-code" id="cb74"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb74-1"><a href="#cb74-1" aria-hidden="true" tabindex="-1"></a><span class="fu">numericq</span>(<span class="fl">1</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="71">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="10986069046129893798" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_10986069046129893798">
|
||
<div style="padding-top: 5px">
|
||
<br>
|
||
<div class="input-group">
|
||
<input id="10986069046129893798" type="number" class="form-control" placeholder="Numeric answer">
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="10986069046129893798_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.getElementById("10986069046129893798").addEventListener("change", function() {
|
||
var correct = (Math.abs(this.value - 1) <= 0);
|
||
var msgBox = document.getElementById('10986069046129893798_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_10986069046129893798")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_10986069046129893798")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-13" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-13">Question</h6>
|
||
<p>Does this function have a limit as <span class="math inline">\(h\)</span> goes to <span class="math inline">\(0\)</span> from the right (that is, assume <span class="math inline">\(h>0\)</span>)?</p>
|
||
<p><span class="math display">\[
|
||
\frac{h^h - 1}{h}
|
||
\]</span></p>
|
||
<div class="cell" data-hold="true" data-execution_count="72">
|
||
<div class="cell-output cell-output-display" data-execution_count="72">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="17232851080111814846" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_17232851080111814846">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_17232851080111814846_1">
|
||
<input class="form-check-input" type="radio" name="radio_17232851080111814846" id="radio_17232851080111814846_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
Yes, the value is <code>-11.5123</code>
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_17232851080111814846_2">
|
||
<input class="form-check-input" type="radio" name="radio_17232851080111814846" id="radio_17232851080111814846_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
Yes, the value is <code>-9.2061</code>
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_17232851080111814846_3">
|
||
<input class="form-check-input" type="radio" name="radio_17232851080111814846" id="radio_17232851080111814846_3" value="3">
|
||
|
||
<span class="label-body px-1">
|
||
No, the value heads to negative infinity
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="17232851080111814846_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_17232851080111814846"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 3;
|
||
var msgBox = document.getElementById('17232851080111814846_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_17232851080111814846")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_17232851080111814846")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-14" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-14">Question</h6>
|
||
<p>Compute the limit</p>
|
||
<p><span class="math display">\[
|
||
\lim_{x \rightarrow 1} \frac{x}{x-1} - \frac{1}{\ln(x)}.
|
||
\]</span></p>
|
||
<div class="cell" data-hold="true" data-execution_count="73">
|
||
<div class="cell-output cell-output-display" data-execution_count="73">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="10000693896182708390" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_10000693896182708390">
|
||
<div style="padding-top: 5px">
|
||
<br>
|
||
<div class="input-group">
|
||
<input id="10000693896182708390" type="number" class="form-control" placeholder="Numeric answer">
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="10000693896182708390_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.getElementById("10000693896182708390").addEventListener("change", function() {
|
||
var correct = (Math.abs(this.value - 0.5) <= 0.001);
|
||
var msgBox = document.getElementById('10000693896182708390_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_10000693896182708390")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_10000693896182708390")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-15" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-15">Question</h6>
|
||
<p>Compute the limit</p>
|
||
<p><span class="math display">\[
|
||
\lim_{x \rightarrow 1/2} \frac{1}{\pi} \frac{\cos(\pi x)}{1 - (2x)^2}.
|
||
\]</span></p>
|
||
<div class="cell" data-hold="true" data-execution_count="74">
|
||
<div class="cell-output cell-output-display" data-execution_count="74">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="4199270321261285223" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_4199270321261285223">
|
||
<div style="padding-top: 5px">
|
||
<br>
|
||
<div class="input-group">
|
||
<input id="4199270321261285223" type="number" class="form-control" placeholder="Numeric answer">
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="4199270321261285223_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.getElementById("4199270321261285223").addEventListener("change", function() {
|
||
var correct = (Math.abs(this.value - 1//4) <= 0.001);
|
||
var msgBox = document.getElementById('4199270321261285223_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_4199270321261285223")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_4199270321261285223")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-16" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-16">Question</h6>
|
||
<p>Some limits involve parameters. For example, suppose we define <code>ex</code> as follows:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="75">
|
||
<div class="sourceCode cell-code" id="cb75"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb75-1"><a href="#cb75-1" aria-hidden="true" tabindex="-1"></a><span class="pp">@syms</span> m<span class="op">::</span><span class="dt">real </span>k<span class="op">::</span><span class="dt">real</span></span>
|
||
<span id="cb75-2"><a href="#cb75-2" aria-hidden="true" tabindex="-1"></a>ex <span class="op">=</span> (<span class="fl">1</span> <span class="op">+</span> k<span class="op">*</span>x)<span class="op">^</span>(m<span class="op">/</span>x)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="75">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
\left(k x + 1\right)^{\frac{m}{x}}
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<p>What is <code>limit(ex, x => 0)</code>?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="76">
|
||
<div class="cell-output cell-output-display" data-execution_count="76">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="11693173723470713795" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_11693173723470713795">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_11693173723470713795_1">
|
||
<input class="form-check-input" type="radio" name="radio_11693173723470713795" id="radio_11693173723470713795_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
\(k/m\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_11693173723470713795_2">
|
||
<input class="form-check-input" type="radio" name="radio_11693173723470713795" id="radio_11693173723470713795_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
\(e^{km}\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_11693173723470713795_3">
|
||
<input class="form-check-input" type="radio" name="radio_11693173723470713795" id="radio_11693173723470713795_3" value="3">
|
||
|
||
<span class="label-body px-1">
|
||
\(e^{k/m}\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_11693173723470713795_4">
|
||
<input class="form-check-input" type="radio" name="radio_11693173723470713795" id="radio_11693173723470713795_4" value="4">
|
||
|
||
<span class="label-body px-1">
|
||
\(0\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_11693173723470713795_5">
|
||
<input class="form-check-input" type="radio" name="radio_11693173723470713795" id="radio_11693173723470713795_5" value="5">
|
||
|
||
<span class="label-body px-1">
|
||
\(m/k\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="11693173723470713795_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_11693173723470713795"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 2;
|
||
var msgBox = document.getElementById('11693173723470713795_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_11693173723470713795")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_11693173723470713795")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-17" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-17">Question</h6>
|
||
<p>For a given <span class="math inline">\(a\)</span>, what is</p>
|
||
<p><span class="math display">\[
|
||
L = \lim_{x \rightarrow 0+} (1 + a\cdot (e^{-x} -1))^{(1/x)}
|
||
\]</span></p>
|
||
<div class="cell" data-hold="true" data-execution_count="77">
|
||
<div class="cell-output cell-output-display" data-execution_count="77">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="1784049165457028926" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_1784049165457028926">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_1784049165457028926_1">
|
||
<input class="form-check-input" type="radio" name="radio_1784049165457028926" id="radio_1784049165457028926_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
\(L\) does not exist
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_1784049165457028926_2">
|
||
<input class="form-check-input" type="radio" name="radio_1784049165457028926" id="radio_1784049165457028926_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
\(e^a\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_1784049165457028926_3">
|
||
<input class="form-check-input" type="radio" name="radio_1784049165457028926" id="radio_1784049165457028926_3" value="3">
|
||
|
||
<span class="label-body px-1">
|
||
\(e^{-a}\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_1784049165457028926_4">
|
||
<input class="form-check-input" type="radio" name="radio_1784049165457028926" id="radio_1784049165457028926_4" value="4">
|
||
|
||
<span class="label-body px-1">
|
||
\(a\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="1784049165457028926_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_1784049165457028926"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 3;
|
||
var msgBox = document.getElementById('1784049165457028926_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_1784049165457028926")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_1784049165457028926")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-18" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-18">Question</h6>
|
||
<p>For positive integers <span class="math inline">\(m\)</span> and <span class="math inline">\(n\)</span> what is</p>
|
||
<p><span class="math display">\[
|
||
\lim_{x \rightarrow 1} \frac{x^{1/m}-1}{x^{1/n}-1}?
|
||
\]</span></p>
|
||
<div class="cell" data-hold="true" data-execution_count="78">
|
||
<div class="cell-output cell-output-display" data-execution_count="78">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="1435370525477231901" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_1435370525477231901">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_1435370525477231901_1">
|
||
<input class="form-check-input" type="radio" name="radio_1435370525477231901" id="radio_1435370525477231901_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
\(n/m\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_1435370525477231901_2">
|
||
<input class="form-check-input" type="radio" name="radio_1435370525477231901" id="radio_1435370525477231901_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
\(m/n\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_1435370525477231901_3">
|
||
<input class="form-check-input" type="radio" name="radio_1435370525477231901" id="radio_1435370525477231901_3" value="3">
|
||
|
||
<span class="label-body px-1">
|
||
\(mn\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_1435370525477231901_4">
|
||
<input class="form-check-input" type="radio" name="radio_1435370525477231901" id="radio_1435370525477231901_4" value="4">
|
||
|
||
<span class="label-body px-1">
|
||
The limit does not exist
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="1435370525477231901_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_1435370525477231901"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 2;
|
||
var msgBox = document.getElementById('1435370525477231901_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_1435370525477231901")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_1435370525477231901")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-19" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-19">Question</h6>
|
||
<p>What does <code>SymPy</code> find for the limit of <code>ex</code> (<code>limit(ex, x => 0)</code>), as defined here:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="79">
|
||
<div class="sourceCode cell-code" id="cb76"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb76-1"><a href="#cb76-1" aria-hidden="true" tabindex="-1"></a><span class="pp">@syms</span> x a</span>
|
||
<span id="cb76-2"><a href="#cb76-2" aria-hidden="true" tabindex="-1"></a>ex <span class="op">=</span> (a<span class="op">^</span>x <span class="op">-</span> <span class="fl">1</span>)<span class="op">/</span>x</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="79">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
\frac{a^{x} - 1}{x}
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<div class="cell" data-hold="true" data-execution_count="80">
|
||
<div class="cell-output cell-output-display" data-execution_count="80">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="18273100106479591421" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_18273100106479591421">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_18273100106479591421_1">
|
||
<input class="form-check-input" type="radio" name="radio_18273100106479591421" id="radio_18273100106479591421_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
\(e^{-a}\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_18273100106479591421_2">
|
||
<input class="form-check-input" type="radio" name="radio_18273100106479591421" id="radio_18273100106479591421_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
\(\log(a)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_18273100106479591421_3">
|
||
<input class="form-check-input" type="radio" name="radio_18273100106479591421" id="radio_18273100106479591421_3" value="3">
|
||
|
||
<span class="label-body px-1">
|
||
\(e^a\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_18273100106479591421_4">
|
||
<input class="form-check-input" type="radio" name="radio_18273100106479591421" id="radio_18273100106479591421_4" value="4">
|
||
|
||
<span class="label-body px-1">
|
||
\(a\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="18273100106479591421_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_18273100106479591421"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 2;
|
||
var msgBox = document.getElementById('18273100106479591421_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_18273100106479591421")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_18273100106479591421")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
<p>Should <code>SymPy</code> have needed an assumption like</p>
|
||
<div class="cell" data-execution_count="81">
|
||
<div class="sourceCode cell-code" id="cb77"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb77-1"><a href="#cb77-1" aria-hidden="true" tabindex="-1"></a><span class="pp">@syms</span> a<span class="op">::</span><span class="dt">postive</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="81">
|
||
<pre><code>(a,)</code></pre>
|
||
</div>
|
||
</div>
|
||
<div class="cell" data-execution_count="82">
|
||
<div class="sourceCode cell-code" id="cb79"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb79-1"><a href="#cb79-1" aria-hidden="true" tabindex="-1"></a><span class="fu">yesnoq</span>(<span class="st">"yes"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="82">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="14337845843244254487" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_14337845843244254487">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_14337845843244254487_1">
|
||
<input class="form-check-input" type="radio" name="radio_14337845843244254487" id="radio_14337845843244254487_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
Yes
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_14337845843244254487_2">
|
||
<input class="form-check-input" type="radio" name="radio_14337845843244254487" id="radio_14337845843244254487_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
No
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="14337845843244254487_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_14337845843244254487"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 1;
|
||
var msgBox = document.getElementById('14337845843244254487_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
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|
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msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
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if (explanation != null) {
|
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explanation.style.display = "block";
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}
|
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})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-the-squeeze-theorem" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-the-squeeze-theorem">Question: The squeeze theorem</h6>
|
||
<p>Let’s look at the function <span class="math inline">\(f(x) = x \sin(1/x)\)</span>. A graph around <span class="math inline">\(0\)</span> can be made with:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="83">
|
||
<div class="sourceCode cell-code" id="cb80"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb80-1"><a href="#cb80-1" aria-hidden="true" tabindex="-1"></a><span class="fu">f</span>(x) <span class="op">=</span> x <span class="op">==</span> <span class="fl">0</span> ? <span class="cn">NaN</span> <span class="op">:</span> x <span class="op">*</span> <span class="fu">sin</span>(<span class="fl">1</span><span class="op">/</span>x)</span>
|
||
<span id="cb80-2"><a href="#cb80-2" aria-hidden="true" tabindex="-1"></a>c, delta <span class="op">=</span> <span class="fl">0</span>, <span class="fl">1</span><span class="op">/</span><span class="fl">4</span></span>
|
||
<span id="cb80-3"><a href="#cb80-3" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(f, c <span class="op">-</span> delta, c <span class="op">+</span> delta)</span>
|
||
<span id="cb80-4"><a href="#cb80-4" aria-hidden="true" tabindex="-1"></a><span class="fu">plot!</span>(abs)</span>
|
||
<span id="cb80-5"><a href="#cb80-5" aria-hidden="true" tabindex="-1"></a><span class="fu">plot!</span>(x <span class="op">-></span> <span class="fu">-abs</span>(x))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="83">
|
||
<p><img src="limits_files/figure-html/cell-84-output-1.svg" class="img-fluid"></p>
|
||
</div>
|
||
</div>
|
||
<p>This graph clearly oscillates near <span class="math inline">\(0\)</span>. To the graph of <span class="math inline">\(f\)</span>, we added graphs of both <span class="math inline">\(g(x) = \lvert x\rvert\)</span> and <span class="math inline">\(h(x) = - \lvert x\rvert\)</span>. From this graph it is easy to see by the “squeeze theorem” that the limit at <span class="math inline">\(x=0\)</span> is <span class="math inline">\(0\)</span>. Why?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="84">
|
||
<div class="cell-output cell-output-display" data-execution_count="84">
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|
||
|
||
<span class="label-body px-1">
|
||
The functions \(g\) and \(h\) squeeze each other as \(g(x) > h(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
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<label class="form-check-label" for="radio_6841378221406852126_2">
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|
||
|
||
<span class="label-body px-1">
|
||
The function \(f\) has no limit - it oscillates too much near \(0\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
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<label class="form-check-label" for="radio_6841378221406852126_3">
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<input class="form-check-input" type="radio" name="radio_6841378221406852126" id="radio_6841378221406852126_3" value="3">
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|
||
<span class="label-body px-1">
|
||
The functions \(g\) and \(h\) both have a limit of \(0\) at \(x=0\) and the function \(f\) is in between both \(g\) and \(h\), so must to have a limit of \(0\).
|
||
</span>
|
||
</label>
|
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</div>
|
||
|
||
|
||
</div>
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</div>
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|
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} else {
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msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
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if (explanation != null) {
|
||
explanation.style.display = "block";
|
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|
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|
||
|
||
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|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
<p>(The <a href="https://en.wikipedia.org/wiki/Squeeze_theorem">Wikipedia</a> entry for the squeeze theorem has this unverified, but colorful detail:</p>
|
||
<blockquote class="blockquote">
|
||
<p>In many languages (e.g. French, German, Italian, Hungarian and Russian), the squeeze theorem is also known as the two policemen (and a drunk) theorem, or some variation thereof. The story is that if two policemen are escorting a drunk prisoner between them, and both officers go to a cell, then (regardless of the path taken, and the fact that the prisoner may be wobbling about between the policemen) the prisoner must also end up in the cell.</p>
|
||
</blockquote>
|
||
</section>
|
||
<section id="question-20" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-20">Question</h6>
|
||
<p>Archimedes, in finding bounds on the value of <span class="math inline">\(\pi\)</span> used <span class="math inline">\(n\)</span>-gons with sides <span class="math inline">\(12, 24, 48,\)</span> and <span class="math inline">\(96\)</span>. This was so the trigonometry involved could be solved exactly for the interior angles (e.g. <span class="math inline">\(n=12\)</span> is an interior angle of <span class="math inline">\(\pi/6\)</span> which has <code>sin</code> and <code>cos</code> computable by simple geometry. See <a href="https://arxiv.org/pdf/2008.07995.pdf">Damini and Abhishek</a>) These exact solutions led to subsequent bounds. A more modern approach to bound the circumference of a circle of radius <span class="math inline">\(r\)</span> using a <span class="math inline">\(n\)</span>-gon with interior angle <span class="math inline">\(\theta\)</span> would be to use the trigonometric functions. An upper bound would be found with (using the triangle with angle <span class="math inline">\(\theta/2\)</span>, opposite side <span class="math inline">\(x\)</span> and adjacent side <span class="math inline">\(r\)</span>:</p>
|
||
<div class="cell" data-execution_count="85">
|
||
<div class="sourceCode cell-code" id="cb81"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb81-1"><a href="#cb81-1" aria-hidden="true" tabindex="-1"></a><span class="pp">@syms</span> theta<span class="op">::</span><span class="dt">real </span>r<span class="op">::</span><span class="dt">real</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="85">
|
||
<pre><code>(theta, r)</code></pre>
|
||
</div>
|
||
</div>
|
||
<div class="cell" data-hold="true" data-execution_count="86">
|
||
<div class="sourceCode cell-code" id="cb83"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb83-1"><a href="#cb83-1" aria-hidden="true" tabindex="-1"></a>x <span class="op">=</span> r <span class="op">*</span> <span class="fu">tan</span>(theta<span class="op">/</span><span class="fl">2</span>)</span>
|
||
<span id="cb83-2"><a href="#cb83-2" aria-hidden="true" tabindex="-1"></a>n <span class="op">=</span> <span class="fl">2</span>PI<span class="op">/</span>theta <span class="co"># using PI to avoid floaing point roundoff in 2pi</span></span>
|
||
<span id="cb83-3"><a href="#cb83-3" aria-hidden="true" tabindex="-1"></a><span class="co"># C < n * 2x</span></span>
|
||
<span id="cb83-4"><a href="#cb83-4" aria-hidden="true" tabindex="-1"></a>upper <span class="op">=</span> n<span class="op">*</span><span class="fl">2</span>x</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="86">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
\frac{4 \pi r \tan{\left(\frac{\theta}{2} \right)}}{\theta}
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<p>A lower bound would use the triangle with angle <span class="math inline">\(\theta/2\)</span>, hypotenuse <span class="math inline">\(r\)</span> and opposite side <span class="math inline">\(x\)</span>:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="87">
|
||
<div class="sourceCode cell-code" id="cb84"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb84-1"><a href="#cb84-1" aria-hidden="true" tabindex="-1"></a>x <span class="op">=</span> <span class="fu">r*sin</span>(theta<span class="op">/</span><span class="fl">2</span>)</span>
|
||
<span id="cb84-2"><a href="#cb84-2" aria-hidden="true" tabindex="-1"></a>n <span class="op">=</span> <span class="fl">2</span>PI<span class="op">/</span>theta</span>
|
||
<span id="cb84-3"><a href="#cb84-3" aria-hidden="true" tabindex="-1"></a><span class="co"># C > n * 2x</span></span>
|
||
<span id="cb84-4"><a href="#cb84-4" aria-hidden="true" tabindex="-1"></a>lower <span class="op">=</span> n<span class="op">*</span><span class="fl">2</span>x</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="87">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
\frac{4 \pi r \sin{\left(\frac{\theta}{2} \right)}}{\theta}
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<p>Using the above, find the limit of <code>upper</code> and <code>lower</code>. Are the two equal and equal to a familiar value?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="88">
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Yes
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<span class="label-body px-1">
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No
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</span>
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</label>
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|
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|
||
<p>(If so, then the squeeze theorem would say that <span class="math inline">\(\pi\)</span> is the common limit.)</p>
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