Squeeze theorem

The squeeze theorem is a useful theorem that lets us compute several types of limits.

Squeeze theorem

If $g(x)\le f(x)\le h(x)$ for all $x\ne a$ in some interval $I$ containing $a$, and ${\lim }_{x\to a}g(x)=l={\lim }_{x\to a}h(x)$, then ${\lim }_{x\to a}f(x)$ exists and:

$$\underset{x\to a}{\lim }f(x)=l$$

Example

Determine whether the following limit exists:

$$\underset{x\to 0}{\lim }{x}^2\sin (\frac{1}{x})$$

We need to build binding functions $g(x)$ and $h(x)$ such that the limits both exist and are equal. We know the following inequalities:

$$-1\le \sin x\le 1$$ $$-1\le \sin (\frac{1}{x})\le 1$$ $$x^2>0$$

Since the second and third inequality is true for all $x\ne 0$, we can multiply the inequalities without changing it.

$$-x^2\le x^2\sin (\frac{1}{x})\le x^2$$

Since the inequality is true for all $x\ne 0$, the first hypothesis is satisfied, and since the limits of all terms are equal to 0, the second hypothesis is also satisfied.

So we conclude that the limit exists and equals 0 by the squeeze theorem.