I put it in exponential form to get $\dfrac{re^{-i \theta}}{re^{i \theta}}$ but I think I'll get $\frac{0}{0}$ which isn't defined and isn't a good enough proof to say it doesn't have a limit.
Find the limit of $\frac{\bar{z}}{z}$ as $z$ goes to $0$.
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sequences-and-series
complex-analysis
limits
convergence-divergence
exponential-function
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0You have to decide how $z$ goes to $0$. In terms of $r$ and $x$ for example. Then you can take the limit as usual. – 2012-11-01
2 Answers
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$z=re^{it}\,\,,\,r>0\,,\,r\in\Bbb R\,\,,\,t\in [0,2\pi]\Longrightarrow\frac{\overline z}{z}=e^{-2it}$
From the above it follows that the limit depends on the angle $\,t\,$ and, thus, it doesn't exist.
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0@DavidThompson For a limit to exist, it has to have the same value, no matter which direction you approach the limit from. This shows that the function approaches a different value depending on $t$. – 2012-11-02
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Consider approaching $0$ from the real axis, then $z \in \mathbb{R}$, so $z = \bar{z}$, hence the limit is $1$.
Now approach from the imaginary axis, now $z = -\bar{z}$ so limit is $-1$. Thus, it does not exist.
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0@DavidThompson Yes, sufficient to show there are subsequences converging to multiple limit points. It's akin to the real sequence $0,1,0,1,0,1,...$ which also diverges for the same reason - 2 subsequences converging to distinct limit points... – 2012-11-01