Answer
$-\infty$,
(the limit does not exist)
Work Step by Step
Since $|x|=-x$ for $x<0$, we have
$|x|=\left\{\begin{array}{lll}
x & if & x \geq 0\\
-x & if & x < 0
\end{array}\right.$
Approaching $x=0$ from the LEFT, means $x < 0$...
$\displaystyle \lim_{x\rightarrow 0-}(\frac{1}{x}-\frac{1}{|x|})=\lim_{x\rightarrow 0-}(\frac{1}{x}-\frac{1}{-x})=\lim_{x\rightarrow 0-}\frac{2}{x}$,
(see sec.1-5, infinite limits)
$\displaystyle \frac{2}{x}$ is negative and increases without bound in magnitude,
so
the limit does not exist
$(\displaystyle \lim_{x\rightarrow 0-}(\frac{1}{x}-\frac{1}{|x|}) =-\infty)$
