Answer
$\lim\limits_{x \to a}[f(x)~g(x)]$ may exist even though neither $\lim\limits_{x \to a}f(x)$ nor $\lim\limits_{x \to a}g(x)$ exist.
Work Step by Step
Let $f(x)$ be defined as follows:
$f(x) = 0~~~$ if $x \lt 0$
$f(x) = 1~~~$ if $x \geq 0$
Let $g(x)$ be defined as follows:
$g(x) = 1~~~$ if $x \lt 0$
$g(x) = 0~~~$ if $x \geq 0$
Then $\lim\limits_{x \to 0}f(x)$ does not exist and $\lim\limits_{x \to 0}g(x)$ does not exist.
However, $f(x)~g(x) = 0$ for all $x$
Therefore, $\lim\limits_{x \to 0}[f(x)~g(x)]$ exists.
$\lim\limits_{x \to a}[f(x)~g(x)]$ may exist even though neither $\lim\limits_{x \to a}f(x)$ nor $\lim\limits_{x \to a}g(x)$ exists.
