Limits part 2: Functions: Techniques
Squeeze lemma
The squeeze lemma also holds for limits of functions.
Squeeze lemma 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.
