Answer
FALSE
Work Step by Step
If the following is true:
$\lim\limits_{x \to 5}f(x)=0$
$\lim\limits_{x \to 5}g(x)=0$
Then it is not true that $\lim\limits_{x \to 5}[\frac{f(x)}{g(x)}]$ doesn't exist.
For instance:
$\lim\limits_{x \to 5}\frac{x-5}{x-5} = \frac{0}{0}$
However, you can simplify the expression: $\lim\limits_{x \to 5}1$
the limit of the expression is 1
Thus, it is FALSE that the limit does not exist.
