Graphs and Limits

In Maple type

plot3d(-x*y/(x^2+y^2), x = -1 .. 1, y = -1 .. 1, grid = [80, 80])

to obtain

Notice that near (x,y)=(0,0) we attain any value between -1/2 and 1/2. (Prove this using limits along lines x=0, y=mx for arbitrary choices of m.)

Now try

plot3d(-x^2*y/(x^4+y^2), x = -1 .. 1, y = -1 .. 1, grid = [80, 80])

to obtain

Imagine tracing on this surface along any line x=0, y=mx; you see that you approach 0. [You should probably check this mathematically to really convince yourself.] However along the parabola y=-x^2 we approach 1/2.