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?

The easiest way is to just compute the derivative separately for $x<0$ and $x>0$. For $x<0$,
$$\lim_{\varepsilon\to 0}\frac{|x+\varepsilon|-|x|}{\varepsilon}=\lim_{\varepsilon\to 0}\frac{-x-\varepsilon+x}{\varepsilon}=-1.$$
For $x>0$,
$$\lim_{\varepsilon\to 0}\frac{|x+\varepsilon|-|x|}{\varepsilon}=\lim_{\varepsilon\to 0}\frac{x+\varepsilon-x}{\varepsilon}=1.$$
