Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: F(z) limit using formal definition

  1. Sep 6, 2010 #1
    1. The problem statement, all variables and given/known data
    For each of the following choices of f(z) use the definition of a limit to obtain lim z-->0 f(z) or prove that the limit doesn't exist
    (a) [tex]\frac{|z|^{2}}{z}[/tex]

    2. Relevant equations
    Formal limit definition

    3. The attempt at a solution
    f(z) = [tex]\frac{|z|^{2}}{z}[/tex]
    f(z) = [tex]\frac{x^{2} + y^{2}}{x +iy}[/tex]
    So if z=(x,0), f(z)=[tex]\frac{x^{2} + 0}{x +i0}[/tex] = x
    Then f(z) --> 0 as (x,y) --> 0 along the real axis
    And if z=(0,y), f(z)=[tex]\frac{0 + y^{2}}{0 +iy}[/tex] = [tex]\frac{y}{i}[/tex]
    Then f(z) --> 0 as (x,y) --> 0 along the imaginary axis
    So maybe lim z-->0 f(z)= 0
    Suppose lim z-->0 f(z)= 0, then for each [tex]\epsilon[/tex] >0 there exists [tex]\delta[/tex] >0 such that 0 < |z - 0| < [tex]\delta[/tex] implies |f(z) - 0| < [tex]\epsilon[/tex]
    Sooooo how do I figure delta out of that? I can't see how to simplify it or what to do next. It's due tomorrow, of course. :smile: And there's a (b) and a (c) , but think I could work them out if I could finish this one.

    Also, can anyone tell me how to make the formulas update? I've changed all the SUP to curly brackets and carrots, but it wont seem to referesh. Edit never mind, seems they just don't show right on preview.
    Last edited: Sep 6, 2010
  2. jcsd
  3. Sep 6, 2010 #2


    User Avatar
    Science Advisor
    Homework Helper

    Here's a suggestion:
    [tex]\left| \frac{z}{|z|} \right| = 1 [/tex]
    So if you write
    [tex] |f(z)| = \left| |z| \cdot \left( \frac{z}{|z|} \right)^{-1} \right| [/tex]
    you can work in your delta.
  4. Sep 6, 2010 #3
    Epsilon= delta. Try to justify it.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook