Answer
See the proof below.
Work Step by Step
One can write $ f(x)$ as follows
$$ f(x)=\frac{f(x)g(x)}{g(x)}.$$
Then, we have
$$\lim\limits_{x \to c}f(x)=\lim\limits_{x \to c}\frac{f(x)g(x)}{g(x)}.$$
We are given that the limit $\lim\limits_{x \to c}f(x)g(x)$ exists and since $\lim\limits_{x \to c}g(x)\neq 0$, then we can apply the quotient law to get the overall limit. Hence, the limit $\lim\limits_{x \to c}f(x)$ exists.
