mathematical physics - Transition probability derivation: How to prove $lim_{alpharightarrowinfty} frac{sin^2alpha x}{alpha x^2} ~=~pidelta(x)$?

How to prove

$$\lim_{\alpha\rightarrow\infty} \frac{\sin^2\alpha x}{\alpha x^2} ~=~\pi\delta(x)~?$$

I have encountered this limit while learning time dependent perturbation and transition probability in Sakurai. How to show this limit? I tried to integrate around $x=0$ but didn't get anything useful? I am out of ideas. Any help is appreciated.