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Homework Help: Limit of f along an indicated curve?

  1. Sep 28, 2011 #1
    I have a problem here with a the syntax of a question.

    1. The problem statement, all variables and given/known data

    "Compute the limit of f as (x,y) approaches (0,0) along the indicated curves.

    f(x,y) = (x2 - y2)/(x2 + y2)

    2. Relevant equations

    a) y = x
    b) y = x/2
    c) y = x2
    d) y = 2x

    3. The attempt at a solution

    I haven't quite started, I've spent most of my time trying to figure out what the conditions under the question meant and how they would change the process of finding the limit.

    If it means anything, I'm expected to use the sequence definition of limits (which I'm capable of doing, I just don't know how to begin).
  2. jcsd
  3. Sep 28, 2011 #2


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    The idea with specifying functions y = g(x) as approach paths to the origin is that you can substitute for y in f(x,y) to reduce the problem to finding a limit of a function of just one variable...
  4. Sep 28, 2011 #3
    So I replace y with the corresponding relation in a-d, right? I'm doing this, so I that I now have f(xn,yn) = (xn2 - (xn2)2)/(xn2 + (xn2)2).
    (for part c)

    Since I now get 0/0, what can I do?
    Last edited: Sep 28, 2011
  5. Sep 28, 2011 #4


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    Do you know about L'Hopital's Rule?
  6. Sep 28, 2011 #5
    Does differentiation work in the same manner with sequences? If they do I'll try that.
  7. Sep 28, 2011 #6


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    You can treat the points xn as discrete points on the curve of a function f(x), but I don't know if they'll fuss at you about not having established this in your course. As an alternative, you can use the "divide numerator and denominator of the rational function by the highest power of x " approach, so you will have a constant leading term "above and below" and the rest are terms which are reciprocals of positive powers of x.

    (I take it this is an analysis course and not just multivariate calculus.)
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