1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

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


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook