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Since xy/(x^2+y^2) > 0 when xy > 0 and is <0 when xy < 0, if the limit is not zero, it does not exist.
As you approach (0,0) along the path x=y, the limit is 1/2
If you approach along y = -x, the limit is -1/2
Looks like there is no limit, since it depends on the approach path.