Question

Use the Squeeze Theorem to determine the value of: $\underset{x\to 0}{lim}{x}^4\sin \frac{\pi }{x}$

Answer 1

We first need to determine lower/upper functions. We’ll start off by acknowledging that provided $x\ne 0$

(which we know it won’t be because we are looking at the limit as $x\to 0$)

we will have,

$-1\le \sin \frac{\pi }{x}\le 1$

Now, simply multiply through this by $x^4$ to get:

$-x^4\le \sin \frac{\pi }{x}\le x^4$

Before proceeding note that we can only do this because we know that $x^4>$ for $x\ne 0$. Recall that if we multiply through an inequality by a negative number we would have had to switch the signs. So, for instance, had we multiplied through by $x^3$ we would have had issues because this is positive if $x>0$ and negative if $x<0$.

Now, let’s get back to the problem. We have a set of lower/upper functions and clearly,

$\underset{x\to 0}{lim}{x}^4=\underset{x\to 0}{lim}-x^4=0$

Therefore, by the Squeeze Theorem we must have,

$\underset{x\to 0}{lim}{x}^4\sin \frac{\pi }{x}=0$.