Single Variable Calculus, Chapter 2, 2.3, Section 2.3, Problem 43

Determine the limit $\lim\limits_{x \rightarrow 0^-} \displaystyle \left(\frac{1}{x} - \frac{1}{|x|}\right)$, if it exists. If the limit does not exist, explain why.


$
\begin{equation}
\begin{aligned}
\lim\limits_{x \rightarrow 0^-} \displaystyle \left(\frac{1}{x} - \frac{1}{|x|}\right) & = \lim\limits_{x \rightarrow 0^-} \left( \frac{1}{x}-\frac{1}{-x}\right)
&& \text{(Theorem of limit of approve values)}\\
\lim\limits_{x \rightarrow 0^-} \displaystyle \left(\frac{1}{x} + \frac{1}{x}\right) & = \lim\limits_{x \rightarrow 0^-} \frac{2}{x} = \frac{2}{0}\\
\end{aligned}
\end{equation}\\
\lim\limits_{x \rightarrow 0^-} \displaystyle \left( \frac{2}{x} \right) \qquad \text{(Does not exist because the denominator will be zero)}
$