Limit of function of two variables

  1. 1. The problem statement, all variables and given/known data
    Fnd the limit, if it exists, or show that the limit does not exist.
    lim [tex]_{(x,y)--> (0,0)} [x^2/(x^2 + y^2)] [/tex]


    2. Relevant equations



    3. The attempt at a solution

    If x = 0, then f(0,y) = 0. f(x,y) --> 0 when (x,y) --> (0,0) along the y-axis.

    If y = 0, then f(x,0) = 1. f(x,y) --> 1 when (,y) --> (0,0) along the x-axis.


    1st, am I doing this right? By simply plugging in y = 0 or x = 0, I can determine the limit?

    2nd, why am I not using L'Hospital's Rule. If (x,y) --> (0,0) then won't the function obviously go to 0/0 which is not real?
     
    Last edited: Mar 16, 2007
  2. jcsd
  3. You are doing it right. An easier way to see what is going on to make the respective polar subsititutions and take the limit as r->0
     
  4. yes, you are correct. Indeed, the limit does not exist, since when taking different paths, the limits are different.
     
  5. HallsofIvy

    HallsofIvy 40,678
    Staff Emeritus
    Science Advisor

    Just to clarify: if you take the limit along two different paths and get different answers then the limit itself does not exist. If you take limits along many different paths and always get the same answer, that does not prove the limit exists because you can't try all possible paths. If you suspect the limit does exist, then the best thing to do is what end3r7 suggested: change to polar coordinates so that the distance from (0,0) is determined by the single variable r. If the limit, as r goes to 0, does not depend on [itex]\theta[/itex] the limit exists.
     
  6. Thanks .
     
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