This is a graph of the surface z = f(x,y) = xy/(x^2+y^2) near the point (0,0).

Remember that in MATLAB, color corresponds to height (red is relatively larger, blue is relatively smaller). On this surface, z = 0 seems to correspond to the green color. Notice that the surface is green approximately over the x and y axes -- we proved in class that the limit along either of these lines (y = 0 or x = 0) will be 0!

The plot of the surface also makes it clear that along the line y = x, the values of the function are approaching 1/2, and along the line y= -x, the values are approaching -1/2 (as (x,y) -> (0,0)).