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Homework Help: Limit of a Function of Two Variables

  1. Feb 1, 2010 #1
    1. The problem statement, all variables and given/known data
    Evaluate the limit of the function f(x,y) = [tex]\frac{y^3}{x^2+y^2}[/tex]

    2. Relevant equations

    3. The attempt at a solution
    Well, I approached this problem using the multiple-path method and found the following:

    [tex]\stackrel{lim}{x\rightarrow 0} [/tex] [tex]\frac{y^3}{x^2+y^2}[/tex] = y
    [tex]\stackrel{lim}{y\rightarrow 0} [/tex] [tex]\frac{y^3}{x^2+y^2}[/tex] = 0

    and am having trouble interpreting these results. I tried doing a polar substitution and found that:

    [tex]\stackrel{lim}{r\rightarrow 0} [/tex] [tex]\frac{y^3}{x^2+y^2}[/tex] = y*sin2(theta)

    My calculus book is very short on the topic, I am pretty much left in the dark. Please bring me into the light. haha.
  2. jcsd
  3. Feb 1, 2010 #2
    Are you trying determine the limit of the function at the origin? Try using the fact that [itex]x^2 + y^2 \geq y^2[/itex] and apply the http://en.wikipedia.org/wiki/Squeeze_theorem" [Broken]
    Last edited by a moderator: May 4, 2017
  4. Feb 2, 2010 #3
    Yes, I am trying to determine the limit as x,y approaches the origin. The problem is that I have no clue how to apply the squeeze theorem. I cant find any good sites on it.
    Last edited by a moderator: May 4, 2017
  5. Feb 2, 2010 #4


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    Homework Helper

    google squeeze theorem - it's also called the sandwich theorem, or the pinching theorem
  6. Feb 2, 2010 #5
    Ahhh okay, so what I would want to say is that [tex]y^2[/tex] [tex]\leq[/tex][tex]\frac{y^3}{x^2+y^2}[/tex][tex]\leq[/tex]x2+y2 ?
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