align fix; theorem style; condition number

quarto/limits/limits_extensions.qmd

@@ -22,6 +22,8 @@ nothing
 
 ---
 
+![To infinity and beyond](figures/buzz-infinity.jpg){width=40%}
+
 
 The limit of a function at $c$ need not exist for one of many different reasons. Some of these reasons can be handled with extensions to the concept of the limit, others are just problematic in terms of limits. This section covers examples of each.
 
@@ -97,8 +99,11 @@ But unlike the previous example, this function *would* have a limit if the defin
 Let's loosen up the language in the definition of a limit to read:
 
 
-> The limit of $f(x)$ as $x$ approaches $c$ is $L$ if for every  neighborhood, $V$, of $L$ there is a neighborhood, $U$, of $c$ for  which $f(x)$ is in $V$ for every $x$ in $U$, except possibly $x=c$.
+::: {.callout-note icon=false}
 
+The limit of $f(x)$ as $x$ approaches $c$ is $L$ if for every  neighborhood, $V$, of $L$ there is a neighborhood, $U$, of $c$ for  which $f(x)$ is in $V$ for every $x$ in $U$, except possibly $x=c$.
+
+:::
 
 
 The $\epsilon-\delta$ definition has $V = (L-\epsilon, L + \epsilon)$ and $U=(c-\delta, c+\delta)$. This is a rewriting of $L-\epsilon < f(x) < L + \epsilon$ as $|f(x) - L| < \epsilon$.
@@ -106,12 +111,16 @@ The $\epsilon-\delta$ definition has $V = (L-\epsilon, L + \epsilon)$ and $U=(c-
 
 Now for the definition:
 
+::: {.callout-note icon=false}
+## The $\epsilon-\delta$ Definition of a right limit
 
-> A function $f(x)$ has a limit on the right of $c$, written $\lim_{x  \rightarrow c+}f(x) = L$ if for every $\epsilon > 0$, there exists a  $\delta > 0$ such that whenever $0 < x - c < \delta$ it holds that  $|f(x) - L| < \epsilon$. That is, $U$ is $(c, c+\delta)$
-
-
+A function $f(x)$ has a limit on the right of $c$, written $\lim_{x  \rightarrow c+}f(x) = L$ if for every $\epsilon > 0$, there exists a  $\delta > 0$ such that whenever $0 < x - c < \delta$ it holds that  $|f(x) - L| < \epsilon$. That is, $U$ is $(c, c+\delta)$
 
 Similarly, a limit on the left is defined where $U=(c-\delta, c)$.
+:::
+
+
+
 
 
 The `SymPy` function `limit` has a keyword argument `dir="+"` or `dir="-"` to request that a one-sided limit be formed. The default is `dir="+"`. Passing `dir="+-"` will compute both one side limits, and throw an error if the two are not equal, in agreement with no limit existing.
@@ -354,7 +363,7 @@ limit(g(x), x=>0, dir="+")
 ## Limits of sequences
 
 
-After all this, we still can't formalize the basic question asked in the introduction to limits: what is the area contained in a parabola. For that we developed a sequence of sums: $s_n = 1/2 \cdot((1/4)^0 + (1/4)^1 + (1/4)^2 + \cdots + (1/4)^n)$. This isn't a function of $x$, but rather depends only on non-negative integer values of $n$. However, the same idea as a limit at infinity can be used to define a limit.
+After all this, we still can't formalize the basic question asked in the introduction to limits: what is the area contained in a parabola. For that we developed a sequence of sums: $s_n = 1/2 \cdot((1/4)^0 + (1/4)^1 + (1/4)^2 + \cdots + (1/4)^n)$. This isn't a function of real $x$, but rather depends only on non-negative integer values of $n$. However, the same idea as a limit at infinity can be used to define a limit.
 
 
 > Let $a_0,a_1, a_2, \dots, a_n, \dots$ be a sequence of values indexed by $n$. We have $\lim_{n \rightarrow \infty} a_n = L$ if for every $\epsilon > 0$ there exists an $M>0$ where if $n > M$ then $|a_n - L| < \epsilon$.