## Calculus (3rd Edition)

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.