Question

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)?

Answer 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.

Answer 2

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

Answer 3

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