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Two Var Limit

  1. Feb 9, 2017 #1
    1. The problem statement, all variables and given/known data

    limit (x -> 0 y -> 0) of xy/sin(x+y)

    2. Relevant equations

    None that come to mind but maybe Lopital's Rule

    3. The attempt at a solution

    I know that the limit does not exist but I am having trouble figuring out how to show that it does not

    using the line x=y gives x^2/sin(2x)
    using the line -x=y gives -x^2/sin(0)

    I am not sure what to do from here or if I am just completely missing what to do
     
  2. jcsd
  3. Feb 9, 2017 #2

    andrewkirk

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    What limit do you get, approaching the origin along that line?

    Now, can you think of a different line along which to approach the origin that gives you a different limit. If you can, you will have proven that the general limit does not exist - because if it did, the limits for all different lines of approach would be the same.

    L'Hopital's rule will be useful in finding the limit for both lines.
     
  4. Feb 9, 2017 #3
    The limit for x^2/sin(2x) is 0
    I cannot think of another line at the moment though
     
  5. Feb 9, 2017 #4

    andrewkirk

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    What are the simplest lines you might try?
     
  6. Feb 9, 2017 #5
    For -x=y it gives -x^2/sin(0) which is undefined I think is there another line that is better though?
     
  7. Feb 9, 2017 #6

    andrewkirk

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    Those are both sloping lines. Try a non-sloping line.
     
  8. Feb 10, 2017 #7

    haruspex

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    Job done?
     
  9. Feb 10, 2017 #8

    andrewkirk

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    Not if one follows the convention that, where the domain of a function is not explicitly stated, it is assumed to be the set of points on which the given formula has a well-defined value. In that case the whole line -x=y is excluded from the domain.
     
  10. Feb 10, 2017 #9

    haruspex

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    Ok, but it still provides the basis for the proof of nonconvergence. Sloping straight lines won't do it. We need a curve which is asymptotically y=-x at (0,0).
     
  11. Feb 10, 2017 #10

    andrewkirk

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    Sloping straight lines won't do it but, if my calcs are correct, a non-sloping line will, given the observation the OP has already made about the limit along the line y=x.
     
  12. Feb 10, 2017 #11

    haruspex

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    Given the symmetry between x and y, any line can be represented by y=ax. The function is ax2/sin(x(1+a)). If a is not -1 then this clearly tends to 0 as x tends to 0.
     
  13. Feb 10, 2017 #12

    andrewkirk

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    @haruspex: My mistake. You're right. Somehow I'd got it into my head that the numerator was x+y rather than xy.
     
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