From 4b7a25496e85ed95e65e551b2b4730b0636c1017 Mon Sep 17 00:00:00 2001 From: arc Date: Tue, 21 Jan 2025 12:24:38 -0700 Subject: [PATCH] vault backup: 2025-01-21 12:24:38 --- education/math/MATH1210 (calc 1)/Limits.md | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/education/math/MATH1210 (calc 1)/Limits.md b/education/math/MATH1210 (calc 1)/Limits.md index 2f30a7c..8573cad 100644 --- a/education/math/MATH1210 (calc 1)/Limits.md +++ b/education/math/MATH1210 (calc 1)/Limits.md @@ -17,7 +17,8 @@ The standard notation for a limit is: $$ \lim_{x \to a} f(x) = L $$ - As $x$ approaches $a$, the output of $f(x)$ draws closer to $L$. In the above notation, $x$ and $a$ are not necessarily equal. - When plotted, the hole is located at $(a, L)$. - +# Indeterminate Limits +If they have a limit of the form $lim_{x \to a} \frac{f(x)}{g(x)}$ where both $f(x) \to 0$ and $g(x) \to 0$ as $x \to a$ then this limit **may or may not** exist # Continuity A function is continuous if their graph can be traced with a pencil without lifting the pencil from the page. @@ -53,7 +54,7 @@ A piece-wise function is *not* considered an elementary function Together, the above theorems tell us that if $a$ is in the domain of an elementary function, then $\lim_{x \to a} f(x) = f(a)$. # Intermediate Value Theorem -Let $f$ be a continuous function on the interval ${} +Let $f$ be a continuous function on the interval $[a, b]$ and let $N$ be any number strictly between $f(a)$ and $f(b)$. Then there exists a number $c$ in $(a, b)$ such that $f(c) = N$. # Definitions