Answer
False.
Work Step by Step
Let us show a counterexample. Let $f(x) = g(x) = x$. We have that $\lim\limits_{x \to 0} f(x) = \lim\limits_{x \to 0} g(x) = 0 $ $\left(\lim\limits_{x \to 0} f(x) \text{ exists} \right)$. But,
$$ \lim\limits_{x \to 0} \frac{f(x)}{g(x)} = \lim\limits_{x \to 0} \frac xx = \lim\limits_{x \to 0} 1 = 1 $$
That is, the limit $\lim\limits_{x \to 0} \frac{f(x)}{g(x)}$ exists.
