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Prove that $$\lim_{z \to 0} \dfrac{\bar{z}}{z}\;\text{ does not exist.}$$

Not sure how to prove this. Any suggestions would be great!

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1  
I feel like I've seen this before, but I'm not finding it.... –  Cameron Buie Nov 4 '12 at 18:26
    
Showing that a limit does not exist is easy. If you can find ways to compute the limit and get different answers, then you are done. –  Braindead Nov 4 '12 at 18:53
    
@Cameron I agree, same here, recently! exact same question...but where? –  amWhy Nov 4 '12 at 19:01

2 Answers 2

Just look at the result you get when you let $z = x+0i$, and let $x$ tend to 0.

Then do the same with $z = 0 + iy$ and let $y$ tend to zero.

For the limit you want, the answers to the above two would have to be the same, but as you find ...

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Consider $z = r e^{i \theta}$, then

$$ \frac{\overline{z}}{z} = e^{-2 i \theta} $$

and this depends on the direction $\theta$.

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