Absolute values in limits

Question

If $f(x)$ is any function, is the limit of $|f(x)|$ at any point equal to the absolute value of the limit of $f(x)$? I can't think of any examples, but I'd like to know if there's a proof for it before using it.

Answer 1

Yes, assuming that $\lim_{x\to a}f(x)$ exists. Then, if $\lim_{x\to a}f(x)=b$, we have $\lim_{x\to a}|f(x)|=|b|$, since$$\bigl||f(x)|-|b|\bigr|\leqslant|f(x)-b|$$for every $x$.