Finding a one sided limit algebraically (not plugging in numbers) 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.

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.