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Math Help - Limit Problem (Multi)

  1. #1
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    Limit Problem (Multi)

    Evaluate the following limit:

    lim([(x,y) -> (0,0)] (xy)/(x^2+y^2)

    The limit of (xy)/(x^2+y^2) as (x,y) approaches (0,0).

    I have no idea.
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  2. #2
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    Quote Originally Posted by fifthrapiers View Post
    Evaluate the following limit:

    lim([(x,y) -> (0,0)] (xy)/(x^2+y^2)

    The limit of (xy)/(x^2+y^2) as (x,y) approaches (0,0).

    I have no idea.
    First, consider the lim along the path x = 0. And thus, we have:

    lim([(0,y) -> (0,0)] 0/(0 + y^2) = lim(y -> 0) 0 = 0

    Then, consider the lim along the path y = 0; thus, we have

    lim([(x,0) -> (0,0)] 0/(x^2 + 0) = lim(x -> 0) 0 = 0

    Although the limits along the above two paths are the same does not necessarily mean that a limit infact exists. In order for a lim to exist, the lim must be the same along every path through (0,0), not just the two picked. Looking at the path y = x, we have:

    lim([(x,x) -> (0,0)] (x*x)/(x^2 + x^2) = lim(x -> 0) x^2/(2x^2) = 1/2

    And thus, since the limit along the above path does not match the first two paths, the limit does not exist.
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  3. #3
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    Thanks, one more limit if you have time please.

    Consider f(x,y)= 2y^2/(x^2-y^2).

    1.) Find the lim along path x = 1. That is, find lim([(1,y)->(1,1)] f(1,y))

    My work: lim(y -> 1) [(-2y^2)/(y^2 - 1)] is undefined

    2.) Find the lim along path y = 1. That is, find lim([(x,1) -> (1,1)] f(x,1))

    Not sure on this bc I think I got 1 wrong.

    3.) What can you conclude from the above results?
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