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How can we show this limit?

  1. Apr 14, 2007 #1
    1. The problem statement, all variables and given/known data
    as x,y goes to 0,0 lim(2x/(x^2+x+y^2)

    3. The attempt at a solution
    It is obvious that the limit is 2 but I can not show it. I tried to change polar coordinates etc. but nothing did not help. What should I do???
  2. jcsd
  3. Apr 14, 2007 #2
    I would suggest doing limits for each varable separately

    make either x=0 or y=0 and do the limits separately, or in other words, the limit in the y direction or x direction.
  4. Apr 14, 2007 #3
    First take the limit as y goes to 0, to get to
    Divide out the common factor of x to get
    Taking this limit will give 2.
  5. Apr 14, 2007 #4

    matt grime

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    You can't do that. the point is that it has to be independent of the path chosen. You cannot let y tend to zero, then x tend to zero - this is just taking a path along the x axis.

    The best advice I heard was to always convert to polars and then let r tend to zero.

    Let t be theta then we have

    2rcos(t)/(r^2+rcos(t)) = 2cos(t)/(r+cos(t))

    As r tends to zero that tends to 2, as long as cos(t) is not zero.

    this of course shows that the limit does not exist, contrary to what the OP asked. But then that was obvious anyway: on the y axis (so x=0) away from y=0 that function is identically zero.
    Last edited: Apr 14, 2007
  6. Apr 14, 2007 #5
    Yes, this is correct, and in the Y direction the limit is negligible because it goes to zero and the limit goes to zero if you ignore the x direction.
  7. Apr 14, 2007 #6


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    OK, if "y goes to 0" first, you get 2x/(x2+ x)= 2/(x+1) which goes to 2 as x goes to 0. If x goes to 0 first, you get 0/y = 0 which goes to 0 as y goes to 0. What does that tell you about the limit itself?
  8. Apr 14, 2007 #7
    corrections: limit does not exist for f(x,y), because limit in the x direction = 2 and in the y = 0.
    Last edited: Apr 14, 2007
  9. Apr 15, 2007 #8
    Quite right, the limit should be the same along all dimensions under consideration or it doesn't exist.
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