Single Variable Calculus, Chapter 2, 2.3, Section 2.3, Problem 58

Give an example that $\lim \limits_{x \to a} [f(x) + g(x)]$ may exist even though neither $\lim \limits_{x \to a} f(x)$ nor $\lim \limits_{x \to a} g(x)$ exists.

Suppose that $\displaystyle f(x) = \frac{2}{x}$ and $\displaystyle g(x) = \frac{-2}{x}$

$\lim \limits_{x \to 0} f(x) \text{ and } \lim \limits_{x \to 0} g(x) \quad \text{ do not exists for the functions are undefined for $$ denominator.}$

$\begin{equation} \begin{aligned} & \text{However,}\\ & \lim \limits_{x \to 0}[f(x) + g(x)] && = \lim \limits_{x \to 0} f(x) \left[\frac{2}{x} + \left( -\frac{2}{x} \right) \right]\\ & \phantom{x} && = \lim \limits_{x \to 0} f(x) \left[\frac{2}{x} -\frac{2}{x} \right]\\ & \phantom{x} && = \lim \limits_{x \to 0} 0\\ & \phantom{x} && = 0 \end{aligned} \end{equation}$