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Calc III - Multivariate limit

  1. Jan 29, 2012 #1
    1. The problem statement, all variables and given/known data
    We define the function [itex]f: \mathbb{R}^2 \to \mathbb{R} [/itex] as

    [itex]
    \begin{equation}
    f(x,y) = \frac{xy^2 ln(x^2 + y^2)}{x^2 + y^2}
    \end{equation}
    [/itex] if [itex](x,y) \neq (0,0)[/itex]. Also note that [itex]f(0,0) = 0[/itex].

    Show that [itex]f[/itex] is continuous at [itex](0,0)[/itex]
    2. Relevant equations



    3. The attempt at a solution
    Polar coords don't work and I don't see a good way to utilize the squeeze theorem which leaves me with delta epsilon. I'm terrible with delta epsilon proofs so I'm wondering if someone can get me started and I'll take it from there.

    Thanks.
     
  2. jcsd
  3. Jan 29, 2012 #2

    SammyS

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    What is the problem with polar coord's?
     
  4. Jan 29, 2012 #3
    I get [itex]ln(r^2)[/itex] which is undefined as [itex]r \to 0[/itex]
     
  5. Jan 29, 2012 #4

    Dick

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    I think polar coordinates will work. Why do you think they don't?
     
  6. Jan 29, 2012 #5
    I figured it out. I feel pretty dumb now.

    It's just like [itex] \displaystyle \lim _{x \to 0} x sin(1/x) = 0[/itex] but [itex]\displaystyle \lim _{x \to 0} sin(1/x)[/itex] is undefined.

    Thanks for the help.
     
  7. Jan 29, 2012 #6

    SammyS

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    Yes.

    I assume you got
    r ln(r2) sin2(θ) cos(θ) .​
    Then took the limit of that as r → 0 .
     
  8. Jan 29, 2012 #7

    Dick

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    I think it's more like a l'Hopital's rule proof. But glad you got it.
     
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