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Determining the limit for function of x and y

  1. Oct 4, 2015 #1
    1. The problem statement, all variables and given/known data
    For f(x,y) = (2x - y^2)/(2x^2 + y), what is the limit as (x,y)->(0,0)?

    YomJXbB.png


    2. Relevant equations


    3. The attempt at a solution
    From this image, it seems that the limit would be non-existent since on one side of the sheet, it goes up and up to infinity whereas from the other side, it plunges down to negative infinity.

    How can I show that the limit is DNE analytically?
     
  2. jcsd
  3. Oct 4, 2015 #2

    andrewkirk

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    The general limit does not exist. However there are limits when one approaches (0,0) from certain directions, eg along the line y=x the limit is 2.

    To prove the general limit does not exist, just pick a convenient direction, ie a line in the x-y plane, and then show that for any ##\delta>0##, two points can be found on the line, both within distance ##\epsilon## from (0,0), for which the values of f(x,y) differ by more than 1. The line y=0 looks promising.
     
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