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Multivariable Limits

  1. May 10, 2009 #1
    1. The problem statement, all variables and given/known data

    Find the limit of:
    (x^2+y^2+2xy)/(x^2+y^2)


    2. Relevant equations
    x = r*cos(theta)
    y= r*sin(theta)

    3. The attempt at a solution

    So what I did was change to polar coordinates. Then it simplifies to:

    (r^2 + 2r^2cos(theta)sin(theta) )/r^2

    Factoring out an r^2 from everything you get:
    1 + 2cos(theta)sin(theta)

    And the limit as r and theta go to zero would appear to be 1. However if I plug in numbers very close to zero on my calculator it tells me that the limit is 2. Little help?
     
  2. jcsd
  3. May 10, 2009 #2

    Dick

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    Your limit depends on theta. That means the limit is different as you approach (0,0) in different ways. E.g. if you approach along the x axis, x=t, y=0 (theta=0), you get 1. If you approach along a line x=t, y=t, (theta=pi/4) you get 2. What does this tell you about the existence of the limit?
     
  4. May 10, 2009 #3
    It doesnt exist? So when i do that method if I find i have thetas in my answer then the limit does not exist?
     
  5. May 10, 2009 #4

    Dick

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    If you get different answers for different values of theta, yes, the limit does not exist.
     
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