$\lim_{z->0}(z)/(|z|)$

Question

$z$ is a complex number. I tried doing this by putting $z=x+iy$ in the equation and then trying to approach the limit by approaching the limit first on real axis and then on real axis.

Answer 1

The limit does not exist. You can tell by looking at the real line. For $x>0$ it gives $1$, and for $x<0$ it gives $-1$.

More generally, you can look at rays of different angles: $z=r e^{i \theta}$, $z/|z|= e^{i\theta}$, which is constant on the ray, and therefore equals the limit on that ray. Thus every ray has a different limit.

Answer 2

$$\lim_{x\to0}\frac{x}{|x|}\ne\lim_{y\to0}\frac{iy}{|iy|}.$$

Answer 3

Hint: $lim_{(x,y)\rightarrow (0,0)} \frac{x+iy}{\sqrt{x^2+y^2}}=\frac{1+im}{\sqrt{1+m^2}}$ along $y=mx$