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    Can some1 explain this to me please?

    For the first one I'm not understanding why they've taken the modulus and how mod(z) is related to (z bar)^2/(z)?
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    ^^^
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    (Original post by e^x)
    ^^^
    Sorry, but what is the question? Are you asking why \ \lim_{z\to 0} \frac{\bar{z}^2}{z} =0 ?
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    (Original post by e^x)
    Can some1 explain this to me please?

    For the first one I'm not understanding why they've taken the modulus and how mod(z) is related to (z bar)^2/(z)?
    For a complex limit to exist, the limit must have the same value, regardless of by which path z \to 0 in the complex plane. Here we have:

    \displaystyle \frac{\bar{z}^2}{z} =\frac{r^2 e^{-2i\theta}}{r e^{i\theta}} = r e^{-3i\theta}

    \displaystyle \frac{\bar{z}}{z} =\frac{r e^{-i\theta}}{r e^{i\theta}} = e^{-2i\theta}

    If z \to 0, how does that distinguish those two cases, and why does one of them fail to have a limit?
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    (Original post by atsruser)
    For a complex limit to exist, the limit must have the same value, regardless of by which path z \to 0 in the complex plane. Here we have:

    \displaystyle \frac{\bar{z}^2}{z} =\frac{r^2 e^{-2i\theta}}{r e^{i\theta}} = r e^{-3i\theta}

    \displaystyle \frac{\bar{z}}{z} =\frac{r e^{-i\theta}}{r e^{i\theta}} = e^{-2i\theta}

    If z \to 0, how does that distinguish those two cases, and why does one of them fail to have a limit?

    For the first case z-->0 is the same as saying r-->0 so the first one has a limit of 0.

    I'm still confused for the second case?
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    (Original post by e^x)
    For the first case z-->0 is the same as saying r-->0 so the first one has a limit of 0.
    Yes. Can you show this more formally? (Hint: Squeeze/Sandwich theorem)

    I'm still confused for the second case?
    As you noted z \to 0 \Rightarrow r \to 0 but \theta can be chosen so that z \to 0 along any radial line. For the limit to exist, the expression must approach the same limiting value regardless of which by radial line z \to 0. The specific radial line is determined by \theta.

    Can you find two different \theta's which give different limits?
 
 
 
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