Answer
Limit does not exist
Work Step by Step
$x\lt 0$ so $|x|=-x$
$\lim\limits_{x \to 0^-}(\frac{1}{x}-\frac{1}{|x|})=\lim\limits_{x \to 0^-}(\frac{1}{x}-\frac{1}{-x})=\lim\limits_{x \to 0^-}\frac{2}{x}=-\infty$
$\therefore$ limit does not exist.
