make pdf file generation work

quarto/limits/limits.qmd

@@ -262,14 +262,14 @@ xs = [1/10^i for i in 1:5]
 This progression can be seen to be increasing. Cauchy, in his treatise, can see this through:
 
 
-$$
+
 \begin{align*}
 (1 + \frac{1}{m})^n &= 1 + \frac{1}{1} + \frac{1}{1\cdot 2}(1 = \frac{1}{m}) +  \\
 & \frac{1}{1\cdot 2\cdot 3}(1 - \frac{1}{m})(1 -  \frac{2}{m}) + \cdots \\
 &+
 \frac{1}{1 \cdot 2 \cdot \cdots \cdot m}(1 - \frac{1}{m}) \cdot \cdots \cdot (1 - \frac{m-1}{m}).
 \end{align*}
-$$
+
 
 These values are clearly increasing as $m$ increases. Cauchy showed the value was bounded between $2$ and $3$ 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:
 
@@ -836,7 +836,7 @@ This accurately shows the limit does not exist mathematically, but `limit(ceil(x
 The `limit` 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 $f$ and $g$ always exist (and are finite) below.
 
 
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+
 \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)
@@ -850,7 +850,7 @@ $$
   \frac{\lim_{x \rightarrow c} f(x)}{\lim_{x \rightarrow c} g(x)}
   &(\text{provided }\lim_{x \rightarrow c} g(x) \neq 0)\\
 \end{align*}
-$$
+
 
 These are verbally described as follows, when the individual limits exist and are finite then: