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Limits of functions with more than one variable

  1. Oct 22, 2007 #1
    1. The problem statement, all variables and given/known data

    (Q) f(x,y) = 2xy/(x^2+y^2), (x,y) not = 0.
    0 (x,y) = 0.
    is continuous at every point except the origin.

    (a) Substitute y=mx and then m=tan(theta) to show that f varies with the line's angle of inclination.

    (b) Use the formula obtained in part (a) to show that the limit of f as (x,y) ---> (0,0) along the line y=mx varies from -1 to 1. depending on the angle of approach.

    2. Relevant equations



    3. The attempt at a solution

    After doing all the substitutions, I get Sin(2theta). From that, it is clear that as the angle changes, so will the limit of f(x,y). Moreover, Sin(2theta) can only have values between -1 and 1. But, how does that mean that the limit of f as (x,y) ---> (0,0) along the line y=mx varies from -1 to 1. depending on the angle of approach.

    Shouldn't it be "the limit of f as (x,y)---> (0,0) ALONG THE CURVE y=Sin(2theta)??
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Oct 23, 2007 #2

    Galileo

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    Science Advisor
    Homework Helper

    You've already fixed the curve before you evaluated the limit.
    You said y=mx, where m=tan(theta). So you're considering the value of f(x,y) for those points on that line and you find that the value doesn't depend on x and y at all, but has the constant value sin(2theta) along that line (minus the origin).
    So clearly the limit along the line varies as you vary theta.
     
  4. Oct 23, 2007 #3
    Thanks!

    Thanks a lot for that little tid-bit!!! :smile:
     
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