calculus - Limit Proof Question

How to prove $\lim_{n\to\infty}\frac{f(x)}{g(x)} = \frac{\lim_{n\to\infty} f(x)}{\lim_{n\to\infty}g(x)}=\frac{L}{M}$ if g(x) is not equal to 0 using $\epsilon-\delta$ definition. I know the proof that uses the idea of $\frac{1}{g(x)}$ and the uses multiplication rule of limit, but I am wondering if there is a direct and more elegant proof.