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Limits in the complex vectorspace

  1. Mar 28, 2012 #1
    Hi,
    I am trying to prove that a limit exists at a point using the epsilon delta definition in the complex plane, but I can't seem to reach a conclusion.
    Here's what I have been trying to get at:

    [itex]\lim_{z\to z_o} z^2+c = {z_o}^2 +c[/itex]

    [itex] |z^2+c-{z_o}^2-c|<\epsilon \ whenever\ 0<|z-z_o|<\delta[/itex]

    [itex]LH=|z^2-{z_o}^2|=|z-z_o||z+z_o|[/itex]

    [itex]=|z-z_o||\overline{z+z_o}|[/itex]

    [itex]=|z-z_o||\bar{z}+\bar{z_o}|[/itex]

    [itex]=|z\bar{z} +z\bar{z_o} -{z_o}\bar{z} -z_o\bar{z_o}|[/itex]

    [itex]=| |z|^2 -|z_o|^2 +2Im(zz_o) |[/itex]

    [itex]\leq||z|^2 -|z_o|^2 +2|z||z_o|| \ (because\ Im(z)\leq|z|)[/itex]

    But I can't get any further. I did this much thinking I could factor it to the square of delta, but that didn't work out because of the positive 2zzo term.If anyone can help me out here, it would be great. Thanks.
     
  2. jcsd
  3. Mar 28, 2012 #2

    tiny-tim

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    hi eptheta! :smile:
    stop there … the |z-zo| part is easy,

    so all you have to do is ensure that |z+zo| is bounded :wink:
     
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