limits - Evaluating and proving $limlimits_{xtoinfty}frac{sin x}x$

I've just started learning about limits. Why can we say

$$\lim_{x\rightarrow \infty} \frac{\sin x}{x} = 0$$ even though $\lim_{x\rightarrow \infty} \sin x$ does not exist?

It seems like the fact that sin is bounded could cause this, but I'd like to see it algebraically.

$$\lim_{x\rightarrow \infty} \frac{\sin x}{x} = \frac{\lim_{x\rightarrow \infty} \sin x} {\lim_{x\rightarrow \infty} x} = ?$$

L'Hopital's rule gives a fraction whose numerator doesn't converge. What is a simple way to proceed here?