Are limits of functions different in the complex plane

Question

I recently learnt about complex numbers and this got me thinking about the limits of functions for example when we say that $\lim\limits_{x \to 4} x^2=16$ we assume that $x^2$ is a function defined for all real numbers and we consider the limit as $x$ approaches $4$ from any direction within the domain of $x^2$ .however why don't we consider the limit as $x$ approaches $4$ from the complex numbers is it because $x^2$ is not defined for complex numbers?

Answer 1

You can think of convergence in $\mathbb C$ as convergence in $\mathbb R^2$ (they have the same topology). So if you take $\lim_{z\to a}f(z)$ where $f(z)$ is a complex function and $a$ is real and this limit exists, then it doesent matter how you aproach $a$ it will have the same limit so if you take the restriction of $f$ to the real line, namely $g=f|_{\mathbb R}$ then $\lim_{z\to a}f(z)=\lim_{x\to a}g(x)$ where the second limit is on the real line.