Define f(x,y) by
Along any line through the origin, f approaches zero. Indeed, any line through
(0,0) may be parametrized by x=ta, y=tb for some constants a and b,
not both zero. As t goes to 0, 
,
which goes to 0 as t goes to zero for any a and b. However, 
does not exist: if you approach the origin along the
parabola 
, you note that 
is constantly 1/2, which doesn't go
to zero.
Here's a picture, where we're looking along the positive x axis towards the
origin. What happens as you approach along any ray which is above the x axis
is that you go up the ridge which is above and then slide down the other
side as you hit the origin. However, if you come in on , you stay on top
of the ridge the whole way, and get a limit of 1/2. (If you come in on a ray
below the x axis, you go down into the trough which is below 
and then
up on the other side until you hit the origin.)
