Calculus

How do I use polar coordinates to prove that the limit of the function f(x,y)=(2xy)/(x^2 +y^2) doesn't exist as it approaches (0,0)?

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  1. well, we know that since xy > 0 if x and y have the same sign, and xy < 0 otherwise, it is clear that the limit does not exist.
    since 2xy = sin2θ, the same applies. sin2θ/r^2 can be either positive or negative, depending on θ, so the limit does not exist.

  2. How did you get 2xy=sin2θ? Should 2xy = 2r^2 cosθsinθ?

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  3. oh, yeah. You are correct. 2xy = r^2 sin2θ
    so, f(x,y) = sin2θ

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