Answer
$\lim\limits_{x \to 0} \frac{\vert 2x-1 \vert-\vert 2x+1 \vert}{x} = -4$
Work Step by Step
When $x \to 0$, then $(2x-1) \lt 0$
Then $\vert 2x-1 \vert = -(2x-1)$
When $x \to 0$, then $(2x+1) \gt 0$
Then $\vert 2x+1 \vert = (2x+1)$
We can evaluate the limit:
$\lim\limits_{x \to 0} \frac{\vert 2x-1 \vert-\vert 2x+1 \vert}{x}$
$= \lim\limits_{x \to 0} \frac{-(2x-1)-(2x+1)}{x}$
$= \lim\limits_{x \to 0} \frac{-4x}{x}$
$= \lim\limits_{x \to 0} -4$
$ = -4$
