Multivariable Epsilon-Delta Limit Definitions

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[5, (ϵ/1.5)] Sin[t], Min[5, (ϵ/1.5)] 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}, 0.1, 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
]

THIS NOTEBOOK IS THE SOURCE CODE FROM

"Multivariable Epsilon-Delta Limit Definitions" from The Wolfram Demonstrations Project
 http://demonstrations.wolfram.com/MultivariableEpsilonDeltaLimitDefinitions/

Contributed by: Spencer Liang (The Harker School)

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