Fascinating limit example, from a simple rational function
Section 14.2
#24, p. 923
Andy Long, Fall, 2015

f[x_,y_]:=x y^3/(x^2+y^6)
width=.01
Plot3D[f[x,y],{x,-width,width},{y,-width,width},AxesLabel->Automatic]
ContourPlot[f[x,y],{x,-width,width},{y,-width,width}]

Plot3D[f[x, y],{x,-width,width},{y,-width,width},MeshFunctions->Function@@@{{{x,y,z},f[x,y]},{{x,y,z},f[x,y]}},MeshStyle->{Orange,Green},PlotLabel->f,Ticks->None]

Manipulate[
Module[
{},
PlotC=Plot[f[c,y],{y,-1,1},PlotRange->1/2]
]
,{c,-1,1}
]

Manipulate[
Module[
{},
PlotC=Plot[f[x,c],{x,-1,1},PlotRange->1/2]
]
,{c,-1,1}
]

Plot[f[x^3,x],{x,-1,1},AxesOrigin->{0,0}]
f[.5^3,.5]

Plot3D[f[x,y],{x,-width,width},{y,-width,width},AxesLabel->Automatic,PlotRange->{-.5,.5}]