Squeeze lemma

Limits part 2: Functions: Techniques

Squeeze lemma

The squeeze lemma also holds for limits of functions.

If $g(x) \leq f(x) \leq h(x)$ and $\lim g(x) = \lim h(x)$ then also $\lim f(x) = \lim g(x) = \lim h(x)$

The $\lim$ s can be replaced by $\lim_{x\to a}$ , $\lim_{x\uparrow a}$ , $\lim_{x\downarrow a}$ , $\lim_{x\to \infty}$ or $\lim_{x\to -\infty}$ . This means there are basically five squeeze lemma's.

You can also still estimate limits. Two very useful inequalities are: $\begin{aligned} -1 \leq \cos(x)&\leq 1 \\[0.25cm] -1 \leq \sin(x) &\leq 1\end{aligned}$

$\lim_{x\to 0} x \sin\left(\dfrac{1}{x}\right) = 0$ Recall that $-1 \leq \sin\left(\frac{1}{x}\right)\leq 1$ for all $x$ . This implies that $x\sin\left(\dfrac{1}{x}\right)$ can be squeezed by the linear functions $y=x$ and $y=-x$ . We know that $\lim_{x\to 0} \pm x = 0$ and therefore the requested limit has to be equal to 0 as well.

Note that this means that the function $f(x) = \left\{ \begin{array}{ll}x \sin\!\left(\dfrac{1}{x}\right) & \text{if }x \neq 0, \\ 0 & \text{if }x=0 \end{array} \right.$ is continuous.