Answer
See the example below.
Work Step by Step
Let $ f(x)=g(x)=\frac{1}{x}$. One can see that $\lim\limits_{x \to 0}f(x)$ and $\lim\limits_{x \to 0}g(x)$ do not exist (because we can not divide by $0$). But the limit $\lim\limits_{x \to 0}(f(x)/g(x))$ does exist:
$$\lim\limits_{x \to 0}(f(x)/g(x))=\lim\limits_{x \to 0}(1)=1.$$
