Along all straight lines the limits are the same, but the limits are not uniform. You can make a plot:
Plot3D[2 x^2 y/(x^4 + y^2), {x, -1, 1}, {y, -1, 1},
PlotPoints -> 200, Exclusions -> Automatic,
MeshFunctions -> {#3 &}]
In every neighbourhood of the origin the function oscillates between the values 1 and -1. It does not converge to zero. Still, along each straight line y == a x
the function goes to zero, although it is not obvious from the picture:
Plot3D[2 x^2 y/(x^4 + y^2), {x, -1, 1}, {y, -1, 1},
PlotPoints -> 200, Exclusions -> Automatic,
MeshFunctions -> {#2/#1 &},
Mesh -> {Tan[Pi/2 Range[-19, 19]/20]}]