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I'm looking for a way to determine a one sided limit algebraically, such as

$\color{blue}{f(x) = \frac {|x|}{x} , x \neq 0}$

I know that you can find the limit by plugging in numbers or graphing it, but there must be a way to find it without using either of those as a crutch.

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    @Tyler - Can you give an example of working that out?2011-05-16

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Recall that $|a| = \begin{cases} a, & \mbox{if } a \ge 0 \\ -a, & \mbox{if } a < 0. \end{cases} $

Using this definition you should be able to use normal limit techniques ($\epsilon-\delta$ or what have you)

Notice, of course, that your limit does not exist as $x$ approaches zero.