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}
$