Sorry, i got confused. That example was for a different purpose: it has a minimum at the origin along all straight lines, but not along parabolas. Try this:
Plot3D[(y/(x^2) )/(1 + (y/x^2)^2), {x, -1, 1}, {y, -1, 1}]
This has no limit at the origin, but
(y/(x^2) )/(1 + (y/x^2)^2) == (x^2 y)/(x^4 + y^2)
so that it has limit 0 along all radiuses. I am in a hurry, I hope I didn't make mistakes...