Single Variable Calculus, Chapter 2, Review Exercises, Section Review Exercises, Problem 29

If $|f(x)| \leq g(x)$ for all $x$, where $\lim \limits_{x \to a} g(x) = 0$. Find $\lim \limits_{x \to a} f(x)$.

Using Squeeze Theorem

$\qquad$ $\lim \limits_{x \to a} - g(x) \leq \lim \limits_{x \to a} |f(x)| \leq \lim \limits_{x \to a} g(x)$

We know that

$\qquad$ $\lim \limits_{x \to a} g(x) = 0$

We have

$\qquad$ $- \lim \limits_{x \to a} g(x) \leq \lim \limits_{x \to a} |f(x)| \leq \lim \limits_{x \to a} g(x)$

Or

$\qquad$ $0 \leq \lim \limits_{x \to a} |f(x)| \leq 0$

So,

$\qquad$ $\lim \limits_{x \to a} f(x) = \lim \limits_{x \to a} g(x)$

Therefore,

$\qquad$ $\lim \limits_{x \to a} f(x) = 0$