How to use definition of limit to compute the derivative of |x| Using definition of limit, I need to show
$$\lim_{\epsilon \to 0} \frac {|x + \epsilon| - |x|}{\epsilon} = \frac {x}{|x| }, x \neq 0$$
How should I proceed to get out of the absolute value signs?

HINT:
Multiply and divide by $$|x+\epsilon|+|x|$$
SPOILER ALERT:

$$\begin{align}\lim_{\epsilon \to 0}\left(\frac{|x+\epsilon|-|x)}{\epsilon}\right)&=\lim_{\epsilon \to 0}\left(\left(\frac{|x+\epsilon|-|x|}{\epsilon}\right)\left(\frac{|x+\epsilon|+|x|}{|x+\epsilon|+|x|}\right)\right)\\\\&=\lim_{\epsilon \to 0}\left(\frac{2\epsilon \,x+\epsilon^2}{\epsilon(|x+\epsilon|+|x|)}\right)\\\\&=\frac{x}{|x|}\end{align}$$

