f[x_, y_]:=(1/390)((x^3)y-(y^3)x)+10;
Manipulate[Show[
(*Plot*)
Plot3D[
f[x, y], {x, -10, 10}, {y, -10, 10} ,PlotStyle->Directive[Opacity[fopacity], Orange]],
 ParametricPlot3D[{Min[1, (ϵ/2)] Sin[t], Min[1, (ϵ/2)] Cos[t], v}, {t, 0,2 π}, {v, 0, 20}, PlotStyle->{Opacity[dopacity], Green},Axes->None,Mesh->None],
 Plot3D[10-ϵ, {x, -10, 10}, {y, -10, 10}, PlotStyle->Directive[Opacity[eopacity], Blue], Mesh->None],
p2 = Plot3D[10+ϵ, {x, -10, 10}, {y, -10, 10}, PlotStyle->Directive[Opacity[eopacity], Blue], Mesh ->None], 
PlotLabel->TraditionalForm[HoldForm[Limit[(1/390)((x^3)y-(y^3)x) + 10,{x,y}->{0,0}] = 10]],
ImageSize->{450,450}
],
(*Controls*)
{{ϵ, 8}, 1, 10, ImageSize->Large,Appearance->"Labeled",ControlPlacement->Top},
"f(x, y) opacity",
{{fopacity,0.9, ""}, 1, 0, ImageSize->Tiny},
Delimiter,
"ϵ opacity",
{{eopacity,0.5, ""}, 1, 0, ImageSize->Tiny},
"δ opacity",
{{dopacity,0.5, ""}, 1, 0, ImageSize->Tiny},SaveDefinitions->True,ControlPlacement->Left
]

Plot3D[(x*y^2)/(x^2+y^4),{x,-2,2},{y,-2,2},BoxRatios->{2,2,1},AxesOrigin->{0,0,0},PlotPoints->200,ImageSize->450]