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The Problem Solver's Resource
 Introduction | Other Tips and Tricks | Methods of Proof | You are currently viewing page 5.

Limits

This section covers limits and some other precalculus topics.

Definition

• $\displaystyle\lim_{x\to n}f(x)$ is the value that $f(x)$ approaches as $x$ approaches $n$.
• $\displaystyle\lim_{x\uparrow n}f(x)$ is the value that $f(x)$ approaches as $x$ approaches $n$ from values of $x$ less than $n$.
• $\displaystyle\lim_{x\downarrow n}f(x)$ is the value that $f(x)$ approaches as $x$ approaches $n$ from values of $x$ more than $n$.
• If $\displaystyle\lim_{x\to n}f(x)=f(n)$, then $f(x)$ is said to be continuous in $n$.

Theorems and Properties

The statement $\lim_{x\to n}f(x)=L$ is equivalent to: given a positive number $\epsilon$, there is a positive number $\gamma$ such that $0<|x-n|<\gamma\Rightarrow |f(x)-L|<\epsilon$.

Let $f$ and $g$ be real functions. Then:

• $\lim(f+g)(x)=\lim f(x)+\lim g(x)$
• $\lim(f-g)(x)=\lim f(x)-\lim g(x)$
• $\lim(f\cdot g)(x)=\lim f(x)\cdot\lim g(x)$
• $\lim(\frac{f}{g})(x)=\frac{\lim f(x)}{\lim g(x)}$

Suppose $f(x)$ is between $g(x)$ and $h(x)$ for all $x$ in the neighborhood of $S$. If $g$ and $h$ approach some common limit L as $x$ approaches $S$, then $\lim_{x\to S}f(x)=L$.