Does the limit of xy/(x^2 + y^2) exist as (x,y)--->(0,0)?

Why or why not??

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.