Limit of $sinleft(frac{pi}{x} right)$ as $x$ approaches $0$ does not exist, with squeeze theorem

I understand that $\frac{\pi}{0}$ would be infinity, hence $\sin\left(\frac{\pi}{x} \right)$ does not have a limit as $x$ approach $0$.

But if I would to use the squeeze theorem:
$$-1 \leq \sin\left(\frac{\pi}{x} \right) \leq 1$$
shouldn't it be zero? Or does the squeeze theorem not work here?