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education/math/MATH1210 (calc 1)/Limits.md
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@ -17,7 +17,8 @@ The standard notation for a limit is:
$$ \lim_{x \to a} f(x) = L $$ $$ \lim_{x \to a} f(x) = L $$
- As $x$ approaches $a$, the output of $f(x)$ draws closer to $L$. In the above notation, $x$ and $a$ are not necessarily equal. - 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 $(a, L)$. - 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 # Continuity
A function is continuous if their graph can be traced with a pencil without lifting the pencil from the page. 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 $\lim_{x \to a} f(x) = f(a)$. 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 # 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 # Definitions