How can you prove that $\lim_{x \to a} |f(x)| = |\lim_{x \to a} f(x)|$?

Question

How do you prove that $\lim_{x \to a} |f(x)| = |\lim_{x \to a} f(x)|$ ?

Answer 1

Randall's answer correctly points out that one can use the continuity of the absolute value function, but it's worth noting that there is a simple direct proof in this case, based on the inequality $||x|-|y||\leq |x-y|$.

Indeed, assuming that $L=\lim_{x\to a}f(x)$ exists, we have $$ ||f(x)|-|L||\leq |f(x)-L|$$ which can be made arbitrarily small by choosing $x$ sufficiently close to $a$.

Answer 2

Absolute value is a continuous function, so it can pass through limits, assuming both exist. (They need not: let $f(x)$ have a jump discontinuity around $a$, where it is $+1$ on the right and $-1$ on the left.)