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$\lim\limits_{x\to 0} |f(x)|$ $\neq$ $\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?

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