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Proving a limit using epsilon/delta

  1. Dec 5, 2009 #1
    1. The problem statement, all variables and given/known data
    Prove that [tex]\mathop {\lim }\limits_{(x,y,z) \to (0,0,0) } \frac{xyz}{{x^2+y^2+z^2 }} = 0[/tex]
    using the epsilon-delta method

    3. The attempt at a solution
    [tex]0<|(x,y,z)-(0,0,0)|=\sqrt{x^2+y^2+z^2}<\delta[/tex]
    Now I have to rewrite:
    [tex]0<\left|\frac{xyz}{{x^2+y^2+z^2 }}-0\right|<\epsilon [/tex]
    So that I find a relationship between epsilon and delta.

    This is where I get stuck... I can't figure out how to do that.

    This is one of my attempts:
    [tex]0<\left|\frac{xyz}{{x^2+y^2+z^2 }}-0\right|\leq \left|\frac{xyz}{{x^2}}\right|=\left|\frac{yz}{{x}}\right| [/tex]

    Any help is very much appreciated!
     
    Last edited: Dec 5, 2009
  2. jcsd
  3. Dec 5, 2009 #2
    Hint: yz<=1/2*(y^2+z^2)
     
  4. Dec 6, 2009 #3
    Then it is easy:

    [tex]0<\left|\frac{xyz}{x^2+y^2+z^2}-0\right|\leq\left|\frac{xyz}{y^2+z^2}\right|\leq\left|\frac{xyz}{2yz}\right|=\left|\frac{x}{2}\right|<\frac{\delta}{2}=\epsilon[/tex]

    How did you come up with: yz<=1/2*(y^2+z^2) ?
    (Why is it true at all?)
     
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