Limit Properties involving complex conjugate

Question

In $\textit{Complex Analysis}$ by Ahlfors, it is mentioned almost immediately after the definition of a limit that $$\lim_{x \to a} f(x) = L \implies \lim_{x \to a} \overline{f(x)} = \overline{L}.$$ However, there is no mention of the properties of the limit: $$\lim_{x \to a} f(\overline{x}).$$ Is there anything to say about this limit given knowledge that $\lim_{x \to a} f(x) = L$?

Answer 1

No, because $\bar{x}$ is a totally different point on the complex plain (unless $x$ is on the real axis). So, one cannot tell the continuity behavior of $f$ around $x$ by knowing the property around $\bar{x}$. For example, if $x=i$, then $\bar{x}=-i$ which is a totally different point and the behavior around $i$ and $-i$ might be totally different.