Manipulate[
Show[
ContourPlot3D[4*x^2+y^2==z,{x,-2,2},{y,-2(Sqrt[2]),2(Sqrt[2])},{z,0,8},BoundaryStyle->None,ContourStyle->{Blue,Opacity[opG]},Mesh->None,AxesOrigin->{0,0,0},ImageSize->600],
(*z=0*)Graphics3D[{Purple,PointSize[0.025],Point[{0,0,0}]}],
ParametricPlot3D[{0.5*Cos[t],Sin[t],1}, {t, 0, 2*Pi}, PlotStyle->{Opacity[b],Thick,Red}],
ParametricPlot3D[{(1/(Sqrt[2]))*Cos[t],(Sqrt[2])*Sin[t],2}, {t, 0, 2*Pi}, PlotStyle->{Opacity[c],Thick,Orange}],
ParametricPlot3D[{Cos[t],2*Sin[t],4}, {t, 0, 2*Pi}, PlotStyle->{Opacity[d],Thick,Green}],
ParametricPlot3D[{(Sqrt[2])*Cos[t],(Sqrt[8])*Sin[t],8}, {t, 0, 2*Pi}, PlotStyle->{Opacity[e],Thick,Cyan}]
],
{{b,0,Text[Style["z=1", RGBColor[0.5,0,0]]]},0,1},{{c,0,Text[Style["z=2", RGBColor[.5,.25,0]]]},0,1},{{d,0,Text[Style["z=4", RGBColor[0,0.5,0]]]},0,1},{{e,0,Text[Style["z=8", RGBColor[.27,.41,.47]]]},0,1},{{opG,0,Text[Style["Graph Opacity", RGBColor[0,0,0.5]]]},0,1},
ControlPlacement->Left
]

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
]