Answer
Does not exist.
Work Step by Step
As $x\to 0^-$, $x\lt0$ we have $|x|=-x$, thus
$$\lim_{x\to 0^-}(\frac{1}{x}-\frac{1}{|x|})=\lim_{x\to 0^-}(\frac{1}{x}+\frac{1}{x})=\lim_{x\to 0^-}(\frac{2}{x})$$
As $x\to 0^-$, $\frac{2}{x}\to -\infty$, the limit does not exist.