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Using the Squeeze Theorem to find Limit

  1. Feb 23, 2010 #1
    1. The problem statement, all variables and given/known data
    lim (x,y) -> (0,0) for:
    (x^3 - y^3) / (x2 + y2)


    2. Relevant equations



    3. The attempt at a solution
    I am checking all the possible lines to check what the limit would be if it did exist:
    y = mx, plug this into the equation above

    (x^3 - (mx)^3) / (x^2 + (mx)^2)

    reduces to:

    x(1 - m^3) / (1 + m^2)

    so, the limit of this when it approaches 0 would be 0 if it exists. Now how do I find out that it does exist?

    I was told to use the squeeze theorem, but I don't know how to find the bounds.
     
  2. jcsd
  3. Feb 23, 2010 #2

    Dick

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    Science Advisor
    Homework Helper

    The usual approach is to use polar coordinates and write x=r*cos(theta) and y=r*sin(theta). Then if (x,y)->(0,0) r must go to zero. Bound the trig part and use the squeeze theorem.
     
  4. Feb 23, 2010 #3
    try to rewrite x^3-y^3=(x-y)(x^2+xy+y^2), and somewhere along the lines use the facts that: x^2=<x^2+y^2, y^2=<x^2+y^2, xy<x^2+y^2, to come up with an upper bound, so to speak, since your lower is 0.

    EDIT: Or what Dick suggested!
     
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