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Finding limits of 2 varible functions

  1. Sep 26, 2009 #1
    1. The problem statement, all variables and given/known data

    Find the limit, if it exist, otherwise show that it doesn't exist.

    lim F
    (x,y) -- > (0,0) ,

    where F = xycos(y) / (3x^2 + y^2)


    I let T(x,y) = F

    Then evaluated the limit as (x,y) - >(x,0) = 0;
    The same goes for the vertical line.

    The answer is it does not exist, But I can't seem to prove it.
     
  2. jcsd
  3. Sep 26, 2009 #2
    Let y = tx where t is an arbitrary non-zero constant. This is approaching the point from a line of finite non-zero slope.
     
  4. Sep 26, 2009 #3
    Let me try :

    F(x,tx) = x(tx)*cos(tx) / (3x^2 + (tx)^2)

    = tx^2*cos(tx) / (3x^2 + t^2*x^2)
    =

    tx^2*cos(tx)
    -------------
    x^2(3 + t^2)

    =

    t*cos(tx)
    -------------
    (3 + t^2)

    I know that cos(tx) <= 1

    so :

    t*cos(tx) t
    ----------- <= -------
    (3 + t^2) 3 + t^2


    not sure what to do next?
     
  5. Sep 26, 2009 #4
    That's it. 1/(3 + t2) is never 0, and approaching along the x or y-axis yields 0. Since the two limits are not equal, the limit does not exist. Recall the definition of the limit.
     
  6. Sep 26, 2009 #5
    OH right, Thanks
     
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