work on limits section

quarto/limits/limits_extensions.qmd

@@ -192,7 +192,7 @@ Describe the limits at $-1$, $0$, and $1$.
 ## Limits at infinity
 
 
-The loose definition of a horizontal asymptote is "a line such that the distance between the curve and the line approaches $0$ as they tend to infinity." This sounds like it should be defined by a limit. The issue is, that the limit would be at $\pm\infty$ and not some finite $c$. This requires the idea of a neighborhood of $c$, $0 < |x-c| < \delta$ to be reworked.
+The loose definition of a horizontal asymptote is "a line such that the distance between the curve and the line approaches $0$ as they tend to infinity." This sounds like it should be defined by a limit. The issue is, that the limit would be at $\pm\infty$ and not some finite $c$. This requires the idea of a neighborhood of $c$, $0 < |x-c| < \delta$, to be reworked.
 
 
 The basic idea for a limit at $+\infty$ is that for any $\epsilon$, there exists an $M$ such that when $x > M$ it must be that $|f(x) - L| < \epsilon$. For a horizontal asymptote, the line would be $y=L$. Similarly a limit at $-\infty$ can be defined with $x < M$ being the condition.
@@ -284,7 +284,7 @@ That is, right limits can be analyzed as limits at $\infty$ or right limits at $
 ## Limits of infinity
 
 
-Vertical asymptotes are nicely defined with horizontal asymptotes by the graph getting close to some line. However, the formal definition of a limit won't be the same. For a vertical asymptote, the value of $f(x)$ heads towards positive or negative infinity, not some finite $L$. As such, a neighborhood like $(L-\epsilon, L+\epsilon)$ will no longer make sense, rather we replace it with an expression like $(M, \infty)$ or $(-\infty, M)$. As in: the limit of $f(x)$ as $x$ approaches $c$ is *infinity* if for every $M > 0$ there exists a $\delta>0$ such that if $0 < |x-c| < \delta$ then $f(x) > M$. Approaching $-\infty$ would conclude with $f(x) < -M$ for all $M>0$.
+Vertical asymptotes are nicely defined with, as with horizontal asymptotes, by the graph getting close to some line. However, the formal definition of a limit won't be the same. For a vertical asymptote, the value of $f(x)$ heads towards positive or negative infinity, not some finite $L$. As such, a neighborhood like $(L-\epsilon, L+\epsilon)$ will no longer make sense, rather we replace it with an expression like $(M, \infty)$ or $(-\infty, M)$. As in: the limit of $f(x)$ as $x$ approaches $c$ is *infinity* if for every $M > 0$ there exists a $\delta>0$ such that if $0 < |x-c| < \delta$ then $f(x) > M$. Approaching $-\infty$ would conclude with $f(x) < -M$ for $M>0$.
 
 
 ##### Examples