# The Sandwich Theorem and a cute little application

The Sandwich Theorem.

Suppose that $g(x) \leq f(x) \leq h(x)$ for all x in some open interval containing c, except possibly at

$x=c$ itself. Suppose also that

$\lim_{x \rightarrow c}g(x) = \lim_{x \rightarrow c}h(x)=L$.

Then, $\lim_{x \rightarrow c}f(x)=L$.

Question:

Prove that for any function f, $\lim_{x \rightarrow c}|f(x)|=0$ implies that

$\lim_{x \rightarrow c}f(x)=0$.

Hint: Use $-|f(x)| \leq f(x) \leq |f(x)|$.