# Can anyone tell me why this limit doesn't exist!

1. Nov 30, 2011

### 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??

2. Nov 30, 2011

### micromass

Staff Emeritus
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!

3. Nov 30, 2011

### blahblah8724

Ah of course! Perfect thank you :)