Complex Analysis Handbook/Functions/Limit

Example:

Show that $\underset{z\to 0}{\lim }\frac{(x+y{)}^2}{x^2+y^2}$ does not exist.
f(z) mush approach the same value along all paths leading to 0, we choose $y=mx$.

$z\to 0\Rightarrow \{\begin{array}{cc}x\to 0& \\ y\to 0& \end{array}$
and for $y=mx;x\to 0⟺y\to 0$
$\underset{z\to 0}{\lim }\frac{(x+y{)}^2}{x^2+y^2}=\underset{x\to 0}{\lim }\frac{x(1+m{)}^2}{x(1+m^2)}=\frac{(1+m{)}^2}{1+m^2}$
It is obvious that the limit depends on $m$