Single Variable Calculus, Chapter 2, 2.3, Section 2.3, Problem 44

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{(Applying the theory of limit for absolute values.)}\\
\lim\limits_{x \rightarrow 0^+} \displaystyle \left(\frac{1}{x} - \frac{1}{x}\right) & = \lim\limits_{x \rightarrow 0^+} 0 && \text{(Evaluating and simplifying)}
\end{aligned}
\end{equation}\\
\boxed{\lim\limits_{x \rightarrow 0^+} \displaystyle \left(\frac{1}{x} - \frac{1}{x}\right) = 0}
$