True/false question: limit of absolute function I have this true/false question that I think is true because I can not really find a counterexample but I find it hard to really prove it. I tried with the regular epsilon/delta definition of a limit but I can't find a closing proof. Anyone that

If $\lim_{x \rightarrow a}  | f(x) | = | A |$ then $ \lim_{x \rightarrow a}f(x) = A  $


The problem is that it isn't true. As a concrete example, $f(x) = x.$ Then $\lim_{x\to a} |f(x)| = a = |-a|$ if $a > 0$ but $\lim_{x\to a} f(x) \not = -a.$ The issue pops up in that $|A|$ can be either $A$ or $-A$ depending on the sign of $A.$
