Answer
See the proof below.
Work Step by Step
Suppose that $\lim\limits_{x \to c}\frac{f(x)}{g(x)}$ exists. Now, we have
$$\lim\limits_{x \to c}f(x)=\lim\limits_{x \to c}g(x)\frac{f(x)}{g(x)}=\lim\limits_{x \to c}g(x)\lim\limits_{x \to c}\frac{f(x)}{g(x)}=0\times\lim\limits_{x \to c}\frac{f(x)}{g(x)}=0\ne L$$
But this contradicts what is given. Consequently, $\lim\limits_{x \to c}\frac{f(x)}{g(x)}$ does not exist.
