I want to find
$$\lim \limits_{x \to 0} \frac{x^2\sin\left(\frac{1}{x}\right)}{\sin(x)}.$$
I know that the numerator and denominator equal to zero if I plug in $x=0$. Can I use L'Hopital's rule even though the limit of $\sin\frac{1}{x}$ does not exist? Because to use L'Hopital's rule both the denominator and the numerator have to be differentiable?