Limit of complex function

• November 14th 2010, 05:04 PM
mukmar
Limit of complex function
Find $\displaystyle\lim_{z \to 0} \frac{\bar{z}^2}{z}$

Attempt: I thought that this limit might not exist since if you let $z = x + iy$, then when approaching from y-axis, the limit equals $iy$ and when approaching from the x-axis, the limit equals $x$.

However the solution says that limit is actually 0. I'm not sure how to get this result.

Any suggestions?
• November 14th 2010, 05:37 PM
FernandoRevilla
Use $|\bar{z}^2/z|=|z|$.

Regards.
• November 14th 2010, 07:30 PM
mukmar
Is there a limit law that says that the limit of the modulus is related to the limit of the function in any way?
• November 14th 2010, 08:09 PM
chisigma
Quote:

Originally Posted by mukmar
Find $\displaystyle\lim_{z \to 0} \frac{\bar{z}^2}{z}$

Attempt: I thought that this limit might not exist since if you let $z = x + iy$, then when approaching from y-axis, the limit equals $iy$ and when approaching from the x-axis, the limit equals $x$.

However the solution says that limit is actually 0. I'm not sure how to get this result.

Any suggestions?

Setting $z=x + i\ y$ You obtain...

$\displaystyle \frac{\bar{z}^{2}}{z} = \frac{x^{2} - y^{2} -2 i x y}{x+i\ y}= \frac{x^{3} -3 x y^{2}}{x^{2}+y^{2}} + i\ \frac{y^{3} -3 x^{2} y}{x^{2}+y^{2}}$ (1)

Now compute separately the $\lim_{(x,y) \rightarrow (0,0)$ of the real and imaginary part of (1)...

Kind regards

$\chi$ $\sigma$
• November 14th 2010, 08:11 PM
tonio
Quote:

Originally Posted by mukmar
Find $\displaystyle\lim_{z \to 0} \frac{\bar{z}^2}{z}$

Attempt: I thought that this limit might not exist since if you let $z = x + iy$, then when approaching from y-axis, the limit equals $iy$ and when approaching from the x-axis, the limit equals $x$.

However the solution says that limit is actually 0. I'm not sure how to get this result.

Any suggestions?

Another idea: write z in polar form, $z=re^{i\phi}\Longrightarrow z\rightarrow 0\Longleftrightarrow r\rightarrow 0\,,\,\overline{z}=re^{-i\phi}$, so:

$\displaystyle{\frac{\overline{z}^2}{z}=\frac{r}{e^ {i\phi}}\xrightarrow [r\to 0]{} 0}$

Tonio
• November 14th 2010, 10:58 PM
FernandoRevilla
Quote:

Originally Posted by mukmar
Is there a limit law that says that the limit of the modulus is related to the limit of the function in any way?

Better:

$|\bar{z}^2/z-0|<\epsilon\Leftrightarrow |z-0|<\epsilon\quad (z\neq 0)$

Now, choose $\delta=\epsilon$.

Regards.