If $\lim _{x \rightarrow a}\left[\frac{f(x)}{g(x)}\right]$ exists, then which one of the following is correct?

(A) Both $\lim _{x \rightarrow a} f(x)$ and $\lim _{x \rightarrow a} g(x)$ must exist.

(B) $\lim _{x \rightarrow a} f(x)$ need not exist but $\lim _{x \rightarrow a} g(x)$ must exist.

(C) Both $\lim _{x \rightarrow a} f(x)$ and $\lim _{x \rightarrow a} g(x)$ need not exist.

(D) $\lim _{x \rightarrow a} f(x)$ must exist but $\lim _{x \rightarrow a} g(x)$ need not exist.