True/false question: limit of absolute function

Question

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 $

Answer 1

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 $A.$

Answer 2

Let $f$ be constant $1$ and $A:=-1$.