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Homework Help: Tough multivariable limit

  1. Mar 18, 2010 #1
    1. The problem statement, all variables and given/known data

    Evaluate this limit,

    lim (x,y) > (0,0) f(x,y)

    where f(x,y) = ((8x+8)(2x+3y)^2) / (sqrt(3x^2 + 14xy + y^2) - sqrt(x^2 -2xy + 8y^2))

    2. Relevant equations


    3. The attempt at a solution

    I figure the first attempt is to rationalize this fraction. But after I rationalized it, it came out as

    f(x,y) = (sqrt(3x^2 + 14xy + y^2) + sqrt(x^2 -2xy + 8y^2)(8x+8)(2x+3y)^2) / (2x^2 + 16xy - 7y^2)

    Which to me is way to complicate it, I tried to approach from (x > y) , (y > x) , (x > 0) , (y > 0) but it does not get simplified at all.
  2. jcsd
  3. Mar 18, 2010 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    OK, so try working with this part:

    (8x+8)(2x+3y)^2) / (2x^2 + 16xy - 7y^2)

    Multiply out the numerator so you get one polynomial divided by another, then carry out the division and see what you get.
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