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?

$$|x|=\begin{cases}x &\text{if $x>0$}\\-x &\text{if $x<0$}\end{cases}$$
So you just need to split into 3 cases: Positive, Negative, and Zero.
For $x=0$, does $\displaystyle\lim_{x\to 0^+}\frac{x}{|x|}=\lim_{x\to0^-}\frac{x}{|x|}?$
I leave the rest to you, as it should be easy.
