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Limit in Multivariable

  1. Sep 17, 2012 #1
    1. The problem statement, all variables and given/known data
    Hi, I have to evaluate the following limit:
    [tex]\lim_{(x,y) \to (1,0)}\frac{x*y-y}{(x-1)^2+y^2}[/tex]


    2. Relevant equations
    I'm pretty sure I have to use the squeeze theorem.


    3. The attempt at a solution
    Well, I'm pretty sure it has something to do with the fact that the top factors like this:
    [tex]\lim_{(x,y) \to (1,0)}\frac{y(x-1)}{(x-1)^2+y^2}[/tex]
    I'm really new to the squeeze theorem so I don't really know how to use it. I believe I have to find some function comparable to this one that is equal to it or greater than it for all values of x and y and one that is equal or less for all values of x and y. Then I have to prove that both have the same limit, so this function must have it as well.
    Oh, and I suspect the limit is 0.
    Can someone give me a hand, please?
     
  2. jcsd
  3. Sep 17, 2012 #2

    LCKurtz

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    I would be looking for an argument that the limit doesn't exist.
     
  4. Sep 17, 2012 #3

    jbunniii

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    You can simplify the problem via a change of variables: let w = x-1. Then the above limit is equivalent to

    [tex]\lim_{(w,y)\rightarrow(0,0)} \frac{wy}{w^2 + y^2}[/tex]

    What happens if you let [itex](w,y) \rightarrow (0,0)[/itex] from different directions?
     
  5. Sep 17, 2012 #4
    Doh! Thank you. I graphed it and it appeared that it did exist, but I see now I graphed the wrong thing.
     
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