Complex Analysis: Prove that the limit exists and is 0

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September 19th 2010, 08:58 AM
joestevens

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.
September 19th 2010, 09:07 AM
Plato

If $|z-0|<1$ then $|z|^3<1$ .
So $|z+z^3-0|\le |z|+|z|^3<2$ .
So make adjustments.
September 19th 2010, 09:48 AM
joestevens

Thanks to Plato (1) is done.

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

$\dfrac{\overline{z}}{z}=\dfrac{\overline{z}^{\,2}} {|z|^{2}}=\dfrac{x^2-y^2}{x^2+y^2}-\dfrac{2xy}{x^2+y^2}i.$

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