Limit of z̄/z as z approaches zero in the complex plane

[blahblah8724]

The limit of the complex function,

$f(z) = \frac{\overline{z}}{z}$ as z tends to zero, where z is in ℂ

This is 1 when z is real and -1 when z is imaginary... but when z is complex??

 

[micromass]

The limit depends crucially from how you approach it. If you approach it vertically then it's -1. When you approach it horizonally, then it's 1.

If you approach it from the line x=y, then it's

$$\lim_{x\rightarrow 0}{\frac{x-ix}{x+ix}}=\frac{1-i}{1+i}$$

If you approach it through a spiral or a parabola then the answer will be something different.

However, for the limit to exist, it wouldn't have to matter how you approach it. Every approach should yield the same answer. This doesn't happen in this case, so the limit does not exist!

 

[blahblah8724]

Ah of course! Perfect thank you :)