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Two-variable calculus

  1. May 3, 2007 #1
    1. The problem statement, all variables and given/known data

    Show that the following limit does not exist:

    lim (x,y) --> (0,0) of x^2 / (y^2 + x^2)

    2. Relevant equations

    3. The attempt at a solution

    I think it involves using l'hospitals rule and using partial derivatives, but I really don't know.
  2. jcsd
  3. May 3, 2007 #2


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    For a limit to exist in multiple dimensions, it must be the same no matter which path you approach the point from. So if (x,y) travels over, say, y=x to (0,0), if the limit exists, it must be the same as if (x,y) travels over y=0 to (0,0).

    So try two paths, show the limit is different depending on how you approach (0,0), and you're done
  4. May 3, 2007 #3
    I'm not really sure how to go about that
  5. May 3, 2007 #4
    I know how to take partial derivatives and directional derivatives...
  6. May 3, 2007 #5
    Set y=x, and see what the limit is when x->0. Then try setting y=0, and see what the limit is as x->0. This is the idea for proving any limit in multiple variables does not exist, just go along different lines, if you get different answers, the limit does not exist.
  7. May 3, 2007 #6
    Ah, I ge tit now, thanks
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