Intuition for the limit of complex functions

Question

We have intuition and somehow geometrical point of view about the limit in the Real functions.
I mean we can think of the limit on the graph of the Real function and imagine how close we can get on the graph to the particular point we want the function limit of.
Is there any similar representation for the Complex functions too?
I'm not looking for bunch of algebraical definitions, signs and symbols for the limit in the Complex functions, what I'm looking for is an intution to understand the very nature of the limit in Complex functions and I'd prefer to have a graphical representation in my mind if there is one.
Thanks

Answer 1

The idea is always the same: One has $$\lim_{x\to\xi} f(x)=\zeta\>,$$ if $f(x)$ is near $\zeta$ for all points $x$ sufficiently near $\xi$ (the exact definition is more involved). Your question is about this idea of "nearness". In ${\mathbb R}$ a point $x$ is near $\xi$ if the distance $|x-\xi|$ is small, say smaller than a certain tolerance $\delta$. The same makes sense in ${\mathbb C}$: A complex number $z$ is near $\zeta\in{\mathbb C}$ if the distance $|z-\zeta|$ is small. This distance $|z-\zeta|$ is nothing other than the euclidean distance between the points $z=x+iy$ and $\zeta=\xi+i\eta$ in the complex plane; so $$|z-\zeta|=\sqrt{(x-\xi)^2+(y-\eta)^2}\ .$$ But one is not obliged to "unpack" $|z-\zeta|$ in this way all the time. In the exact definition of limit one just writes $|z-\zeta|<\epsilon$ to indicate that the point $z$ is less than $\epsilon$ away from $\zeta$.