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Homework Help: Show the limit exists or does not exist

  1. Nov 3, 2011 #1
    1. The problem statement, all variables and given/known data
    c2f1f923-74de-46af-bb68-3ccd4efeb33a.jpe

    3. The attempt at a solution
    it does not exist, but i cant get different limits, im only getting it approaches 0

    i replaced x=1 only, y=0 only, y=x, x=y but i'm only getting 0 as my limit.

    thanks
     
    Last edited: Nov 3, 2011
  2. jcsd
  3. Nov 3, 2011 #2

    mathwonk

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    this is a trick. notice it is equivalent to finding the limit of [xy]/(x^2+y^2) as (x,y)-->(0,0).

    this one is very familiar, and is done by letting x=y.
     
  4. Nov 3, 2011 #3
    im still approaching 0 when x=y
     
  5. Nov 3, 2011 #4

    gb7nash

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    You shouldn't be. Replacing y with x:

    [tex]\lim_{(x,y) \to (0,0)} \frac{x*x}{x^2+x^2} = ...[/tex]
     
  6. Nov 3, 2011 #5
    oh so first i rewrite the limit into a more familiar one and then solve. but i don't see how they are related.

    edit:nvm i got it.
     
    Last edited: Nov 3, 2011
  7. Nov 4, 2011 #6

    SammyS

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    Yes, if you let x=1, and take the limit of the result as y→0, the limit is zero.

    Similarly, letting y=0 and taking the limit of the result as x→1, also gives zero.

    Use these results along with the limit obtained as others have suggested. What's your final conclusion?
     
  8. Nov 4, 2011 #7
    i showed the relation between the original limit and [xy]/(x^2+y^2) as (x,y)-->(0,0).
    numerator factorizes to y(x-1), then i replaced (x-1) with t, and as x→1, t→0. new limit does not exist because it approaches 1/2 and 0.
     
  9. Nov 4, 2011 #8

    SammyS

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    Correct !
     
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