Evaluating $\lim_{x\to 0}x^4\sin\left(\frac{\pi}{3x^2}\right)$. The answer is $0$, but plugging in $x=0$ makes it undefined.

Question

$$\lim_{x\to 0}x^4\sin\left(\frac{\pi}{3x^2}\right)$$

The answer is $0$, but when I plug in the limit to this question, the denominator will be $0$, which will make this undefined. So, I don't get how does this limit equal to $0$.

Please help, thank you!

Answer 1

Recall that when we consider $x\to 0$ we never reach the value $x=0$ therefore the expression is always well defined, moreover we have $|\sin \theta|\le 1$ and from here we can easily conclude by squeeze theorem using that

$$\left|x^4\sin\left(\frac{\pi}{3x^2}\right)\right| \le x^4 \cdot 1=x^4$$

Answer 2

Note that $\sin\left(\frac{\pi}{3x^2}\right)$ is a bounded function and $\lim_{x \to 0} x^4=0$. By a theorem in Real Analysis, this limit must be equal to $0$.