Existence of Limits of complex functions

Question

I just started to study complex variables and had a problem in trying to show that the limit does not exist. $F(z) =\frac{(x +y) ^2}{x^2 + y^2}$ as $z$ approaches $0$.

I was getting $1$, which to me showed that the limit existed.

Answer 1

Here's some visual aid, courtesy of Mathematica, to supplement Sobi's answer:

As we can see, the function looks terrible, and its value at $0$ is not defined.

Answer 2

The issue here is that the limit should exist and be equal along any curve that converges to the origin. So for instance, if you take the curve $x=t, y=0$ (i.e. if you approach the origin along the real axis), you find that $$\lim_{z\to 0}\frac{(x +y) ^2}{x^2 + y^2} = \lim_{t\to 0}\frac{(0+t)^2}{0^2+t^2} = \lim_{t\to 0}\frac{t^2}{t^2} = 1.$$ What happens if you instead take the curve $x=t, y=t$ (i.e. if you approach the origin at a 45 degree angle)?