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Find the Limit of Multivariable Function

  1. Feb 22, 2012 #1
    1. The problem statement, all variables and given/known data
    Find the limit of:
    lim (x,y) ->(0,0) (x^2*y)/(x^2 + y^4)


    2. Relevant equations
    x=rcosθ
    y=rsinθ


    3. The attempt at a solution
    lim r->0 for all steps
    L = (rcosθ)^2*(rsinθ)/[(rcosθ)^2 + (rsinθ)^4]
    L = r^3 (cosθ)^2 (sinθ) / [ r^2 * (cosθ)^2 + r^4 * (sinθ)^4]
    L = r (cos^2 * sin ) / (cos^2 + r^2*sin^4)

    That's as far as I can get. I thought about trying to use r^2 in the denominator to work back around to a sin^2 + cos^2 or trying to convert the terms in the denominators into r and either cos or sin so I could get rid of one of the terms and nothing seems to be working.

    Oh, I should mention that I tried a couple of different paths and my limit seemed to always equal 0. I know that I cannot prove a limit exists with this method, I can only prove that the limit does not exist. I'm fairly certain the limit = 0, but I can't figure out a way to determine it.

    This is a problem that I'm supposed to be thinking about for discussion on Thursday. So I don't want the answer, but guidance. Thanks!!

    -r
     
  2. jcsd
  3. Feb 22, 2012 #2
    try using the fact x^2+y^4>=x^2
     
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