I've come across the following transformation:

$$\frac{1}{2}\lim\_{x \to 0}\frac{x}{\sin x}=\frac{1}{2}\frac{1}{\lim\_{x \to 0}\frac{\sin{x}}{x}}$$

But I can't quite understand why and how it works. I would be grateful if someone explained why it's correct.

Thanks!