Squeeze Theorem

If a function is trapped between two others that share the same limit, it shares that limit too.

If $g(x) \le f(x) \le h(x)$ near $a$ (except possibly at $a$ ) and $\lim_{x\to a} g(x) = \lim_{x\to a} h(x) = L$ , then $\lim_{x\to a} f(x) = L$ as well — $f$ is "squeezed" between two functions converging to the same value.

Examples

$\displaystyle\lim_{x\to0} x^2\sin(1/x) = 0$ since $-x^2 \le x^2\sin(1/x) \le x^2$ and both bounds go to $0$ .