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.


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