68 lines
3.1 KiB
Markdown
68 lines
3.1 KiB
Markdown
# Introduction
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Every mathematical function can be thought of as a set of ordered pairs, or an input value and an output value.
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- Examples include $f(x) = x^2 + 2x + 1$, and $\{(1, 3), (2, 5), (4, 7)\}$.
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**A limit describes how a function behaves *near* a point, rather than *at* that point.***
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- As an example, given a *well behaved function* $f(x)$ and the fact that:
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- $f(1.9) = 8.41$
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- $f(1.999) = 8.99401$
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- $f(2.1) = 9.61$
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- $f(2.01) = 9.061$
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- $f(2.0001) = 9.0006$
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We can note that the smaller the distance of the input value $x$ to $2$, the smaller the distance of the output to $9$. This is most commonly described in the terms "As $x$ approaches $2$, $f(x)$ approaches $9$", or "As $x \to 2$, $f(x) \to 9$."
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Limits are valuable because they can be used to describe a point on a graph, even if that point is not present.
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# Standard Notation
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The standard notation for a limit is:
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$$ \lim_{x \to a} f(x) = L $$
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- As $x$ approaches $a$, the output of $f(x)$ draws closer to $L$. In the above notation, $x$ and $a$ are not necessarily equal.
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- When plotted, the hole is located at $(a, L)$.
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# Indeterminate Limits
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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
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# Continuity
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A function is continuous if their graph can be traced with a pencil without lifting the pencil from the page.
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Formally, a function $f$ is continuous at a point $a$ if:
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- $f(a)$ is defined
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- $\lim_{x \to a} f(x)$ exists
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- $\lim_{x \to a} = f(a)$
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- A function is continuous on the open interval $(a, b)$ if it is continuous at all points between $a$ and $b$
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- A function is continuous on the closed interval $[a, b]$ if it is continuous at all points between $a$ and $b$
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# Elementary Functions
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An elementary function is any function that is defined using:
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- Polynomial functions
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- Rational functions
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- Root functions
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- Trig functions
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- Inverse trig functions
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- Exponential functions
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- Logarithmic functions
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- Operations of:
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- Addition
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- Subtraction
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- Multiplication
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- Division
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- Composition
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A piece-wise function is *not* considered an elementary function
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- If $f$ and $g$ are continuous at a point $x = a$ and $c$ is a constant then the following functions are also continuous at $x = a$
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- If $g$ is continuous at $a$ and $f$ is continuous at $g(a)$, then $f(g(a))$ is continuous at $a$
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- If $f$ is an elementary function and if $a$ is in the domain of $f$, then $f$ is continuous at $a$
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Together, the above theorems tell us that if $a$ is in the domain of an elementary function, then $\lim_{x \to a} f(x) = f(a)$.
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# Intermediate Value Theorem
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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 $f(c) = N$.
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# Definitions
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| Term | Definition |
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| --------------------- | ----------------------------------------------------------------------------- |
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| Well behaved function | A function that is continuous, has a single value, and is defined everywhere. |
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