Math 22 Limits

The Limit of a Function

 Definition of the Limit of a Function
 If $f(x)$ becomes arbitrarily close to a single number $L$ as $x$ approaches $c$ from either side, then
 $\lim _{x\to c}f(x)=L$
 which is read as "the limit of $f(x)$ as $x$ approaches Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c}
 is $L$

Note: Many times the limit of $f(x)$ as $x$ approaches $c$ is simply $f(c)$, so limit can be evaluate by direct substitution as $\lim _{x\to c}f(x)=f(c)$

Properties of Limits

Let $b$ and $c$ be real numbers, let $n$ be a positive integer, and let $f$ and $g$ be functions with the following limits $\lim _{x\to c}f(x)=L$ and $\lim _{x\to c}g(x)=K$. Then

1. Scalar multiple: $\lim _{x\to c}[bf(x)]=bL$ 2. Sum or difference: $\lim _{x\to c}[f(x)\pm g(x)]=L\pm K$

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