1. Not finding help here? Sign up for a free 30min 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!

Help me with multivariable limits

  1. Feb 13, 2015 #1

    RJLiberator

    User Avatar
    Gold Member

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

    Lim (x,y,)--> (1, 3) of (x^2-1)/(xy-y)

    2. Relevant equations

    I know that the answer is 2/3 according to wolfram alpha multivariable limit calculator.

    3. The attempt at a solution

    So this is my first time doing multivariable limits, I've studied the following:

    1) Direct substitution : Well, this doesn't work in our case as we get 0/0.
    2) Try to go along different paths: say y=x and so forth. This also doesn't work. If we make y=x then we get 0/0. And I also don't feel good about trying to prove that the limit exists with this path idea.
    3) The 'squeeze' idea: I'm not really sure how to apply this to this particular problem. Is this what I need to do?

    Is there anything else that I am not seeing?
     
    Last edited: Feb 13, 2015
  2. jcsd
  3. Feb 13, 2015 #2

    wabbit

    User Avatar
    Gold Member

    One way to get past a 0/0 answer is to find if it can be rewritten as 0*a/0*b,and then see if the 0 part can be taken out. Have you tried that?
     
  4. Feb 13, 2015 #3

    RJLiberator

    User Avatar
    Gold Member

    No, I have not.

    So 0*a/0*b, hm.
    Do you mean such as setting x=0 where it becomes
    (0)^2-1/((0)*y)-y)
    And then we get the answer of -1/-3 which is 1/3?

    This seems to work out well! However, is this enough to prove that the limit does indeed exist?
     
  5. Feb 13, 2015 #4

    wabbit

    User Avatar
    Gold Member

    Hmm no, not quite, why set x=0?
    No, by "0" here I mean for instance " the value of xy - y at (x=1,y=3). Could that turn out to be a product?
     
  6. Feb 13, 2015 #5

    RJLiberator

    User Avatar
    Gold Member

    I was setting x=0 to go along that path? :/

    xy-y = 1(3)-(3) = 0 ?
    y(x-1) = 3(1-1) = 0?

    I'm not sure what you mean by turn out to be a product :eek:.
     
  7. Feb 13, 2015 #6

    wabbit

    User Avatar
    Gold Member

    You're on the right path.
     
  8. Feb 13, 2015 #7

    RJLiberator

    User Avatar
    Gold Member

    AHHHHHH. I was thinking way too much.
    Factor the denominator and numerator.
    Simplifies to
    (x+1)/3
    plug in values, and walouh, 2/3 is the answer I was looking for.

    BEAUTIFUL. Thank you for your guidance.
     
  9. Feb 13, 2015 #8

    wabbit

    User Avatar
    Gold Member

    Right. It's a little better if you factor to (x+1)/y and only substitute at the end because then you can see the result is true whichever way (x,y) goes to (1,3), while you only really proved it if y goes to 3 first, then x goes to 1. But you've got it now.
     
  10. Feb 13, 2015 #9

    RJLiberator

    User Avatar
    Gold Member

    Indeed, excellent rigorous observation.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Help me with multivariable limits
Loading...