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

  1. September 19th 2010, 08:58 AM #1
    joestevens
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    Question Complex Analysis: 1-Prove that the lim exists & =0 and 2-Prove that the limit DNE

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

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

    The only thing I can use the \epsilon-\delta definition.
    I have a few like each of these to do, but no examples were given.
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  2. September 19th 2010, 09:07 AM #2
    Plato
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    If |z-0|<1 then |z|^3<1.
    So |z+z^3-0|\le |z|+|z|^3<2.
    So make adjustments.
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  3. September 19th 2010, 09:48 AM #3
    joestevens
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    Thanks to Plato (1) is done.


    I still need help on (2).
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  4. September 19th 2010, 10:24 AM #4
    Plato
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    \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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