Are we always allowed to assume that $\lim_{x \rightarrow a} \lvert f(x)\rvert = \lvert\lim_{x \rightarrow a} f(x)\rvert$?
I don't think so, but I can't find a counterexample.
2026-03-25 07:44:36.1774424676
$\lim_{x \rightarrow a} \lvert f(x)\rvert = \lvert\lim_{x \rightarrow a} f(x)\rvert$?
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Take$$f(x)=\begin{cases}1&\text{ if }x\geqslant 0\\-1&\text{ otherwise.}\end{cases}$$Then $\lim_{x\to0}\bigl|f(x)\bigr|=1$, whereas $\lim_{x\to0}f(x)$ doesn't exist.