Results 1 to 6 of 6

Thread: Limit of complex function

  1. #1
    Newbie mukmar's Avatar
    Joined
    Nov 2010
    From
    Right behind you
    Posts
    20

    Limit of complex function

    Find $\displaystyle \displaystyle\lim_{z \to 0} \frac{\bar{z}^2}{z}$

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

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

    Any suggestions?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,163
    Thanks
    46
    Use $\displaystyle |\bar{z}^2/z|=|z|$.

    Regards.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie mukmar's Avatar
    Joined
    Nov 2010
    From
    Right behind you
    Posts
    20
    Is there a limit law that says that the limit of the modulus is related to the limit of the function in any way?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    6
    Quote Originally Posted by mukmar View Post
    Find $\displaystyle \displaystyle\lim_{z \to 0} \frac{\bar{z}^2}{z}$

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

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

    Any suggestions?
    Setting $\displaystyle z=x + i\ y$ You obtain...

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

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    3
    Quote Originally Posted by mukmar View Post
    Find $\displaystyle \displaystyle\lim_{z \to 0} \frac{\bar{z}^2}{z}$

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

    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, $\displaystyle z=re^{i\phi}\Longrightarrow z\rightarrow 0\Longleftrightarrow r\rightarrow 0\,,\,\overline{z}=re^{-i\phi}$, so:

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

    Tonio
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,163
    Thanks
    46
    Quote Originally Posted by mukmar View Post
    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:

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

    Now, choose $\displaystyle \delta=\epsilon$.

    Regards.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. limit from sides in complex function..
    Posted in the Calculus Forum
    Replies: 14
    Last Post: Jul 12th 2010, 01:16 PM
  2. Limit of a complex function
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Dec 14th 2009, 05:38 PM
  3. Limit of complex function
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: Oct 7th 2009, 07:55 PM
  4. Limit, Limit Superior, and Limit Inferior of a function
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Sep 3rd 2009, 05:05 PM
  5. Replies: 1
    Last Post: Mar 3rd 2008, 07:17 AM

Search Tags


/mathhelpforum @mathhelpforum