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Homework Help: Multivariable limits

  1. Oct 9, 2009 #1
    1. The problem statement, all variables and given/known data

    Verify the following limit by using delta-epsilon arguments

    2. Relevant equations

    lim (x, y) -> (1, -1) of xy^2 = 1

    3. The attempt at a solution

    Right, so I'm having some trouble with these delta-epsilon proofs for multivariable limits. Some of them are easier than others; I'm talking mainly about cases where the limit = 0, when the limit is some other constant like in the above question I'm not sure how to simplify it and try to get an answer.

    I'm aware of some of the basic ideas here, like:

    sqrt(x^2 + y^2) < delta => |f(x,y) - L| < epsilon

    This also implies that |x| < delta and |y| < delta, which seems to be what you use in practice to solve most of these things, rather than the above definition.

    If I try that on the above equation though, I get |delta^3 - 1| < epsilon. What on earth do I do from here? How do I deal with the constant in situations like these?
  2. jcsd
  3. Oct 9, 2009 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Hint: Try writing |xy2 -1| = |(xy2 - y2) + (y2 - 1)| to get started.
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