finding limit of a complex function using polar coordinates

Question

The limit of $(x+iy)/(x-iy)$ as (x,y) approaches (0,0) does not exist because if we choose the path y = mx, we get that the limit depends on path. But if I consider that function as $Z/Z̅$, ( where $Z̅$ denote the conjugate of Z) I can write it as $exp(i2θ)$. So here as (r,θ) approaches (0,0), i.e. as θ goes to zero why we can't say limit approaches 1. The graph of exp(Z) when ReZ = 0 will be a unit circle, but then how to evaluate limit at origin. I'm little confused. So can someone give a clarity on how to evaluate the limit of the function $Z/Z̅$ as Z approaches zero using polar coordinates. Thanks in advance.

Answer 1

In polar coordinates, approaching the origin means $r$ approaches $0$, with $\theta$ unrestricted.

In particular, $\theta$ need not approach $0$.

Thus, as $r$ approaches $0$, $\exp(2i\theta)$ does not approach a limit, since $\theta$ is free to roam.