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Thread: Complex Analysis: Prove that the limit exists and is 0

  1. #1
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    Question Complex Analysis: 1-Prove that the lim exists & =0 and 2-Prove that the limit DNE

    1) Prove that $\displaystyle \lim_{z\to0}z+|z|^3$ exists and equals $\displaystyle 0$.

    2) Prove by letting $\displaystyle z$ approach $\displaystyle 0$ along suitable rays that $\displaystyle \lim_{z\to0}{\bar{z}\over z}$ fails to exist.

    The only thing I can use the $\displaystyle \epsilon-\delta$ definition.
    I have a few like each of these to do, but no examples were given.
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  2. #2
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    If $\displaystyle |z-0|<1$ then $\displaystyle |z|^3<1$.
    So $\displaystyle |z+z^3-0|\le |z|+|z|^3<2$.
    So make adjustments.
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  3. #3
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    Thanks to Plato (1) is done.


    I still need help on (2).
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  4. #4
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    $\displaystyle \dfrac{\overline{z}}{z}=\dfrac{\overline{z}^{\,2}} {|z|^{2}}=\dfrac{x^2-y^2}{x^2+y^2}-\dfrac{2xy}{x^2+y^2}i.$
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