Calculus: Early Transcendentals 8th Edition

Let $a = 0, f(x) = \frac{1}{x}, g(x) = -\frac{1}{x}$. Neither of the two individual limits exist, but since $f(x) + g(x) = 0$, we have $\lim_{x \to a}[f(x) + g(x)]$ exists and is 0.