$\lim\limits_{x\to 0} |f(x)|$ $\neq$ $\infty $ and if $\lim\limits_{x\to 0} f(x)$ exists, it's $= 0$.
Prove that there is a function $g$ such that $\lim\limits_{x\to 0} g(x)$ does not exist, but $\lim\limits_{x\to 0} f(x)g(x)$ does exist.
Or is this not true?