Answer
False.
Work Step by Step
Let $f(x) = x, \, g(x) = x^2$. $\lim\limits_{x \to 0} f(x) = \lim\limits_{x \to 0} g(x) = \infty$. But,
$$ \lim\limits_{x \to 0} \frac{f(x)}{g(x)} = \lim\limits_{x \to 0} \frac {x}{x^2} = \lim\limits_{x \to 0} \frac 1x = \infty $$
That is, $\lim\limits_{x \to 0} \frac{f(x)}{g(x)} \ne 1$.
