MTH 212: Limits (Section 13.2)

Example I: A “pizza slice” function that has no limit as (x,y) goes to (0,0).

h[x_,y_]:=Piecewise[{{1,x>0 && y >0},{0,True}}]
Plot3D[h[x,y],{x,-5,5},{y,-5,5},Mesh -> None, AxesLabel->{"x","y","z"}]

ContourPlot[h[x,y],{x,-5,5},{y,-5,5},ContourLabels->True]

Show[
Plot3D[h[x,y],{x,-5,5},{y,-5,5},Mesh -> None, AxesLabel->{"x","y","z"}],
ParametricPlot3D[{t,t,h[t,t]},{t,0,5},PlotStyle->{Dashed}],
ParametricPlot3D[{t,t,h[t,t]},{t,-5,0},PlotStyle->{Red,Dashed}],
ParametricPlot3D[{t,-t,h[t,-t]},{t,-5,5},PlotStyle->{Red,Dashed}]
]

Example II : Plots of a function designed to show that the function has no limit as (x,y) approaches (0,0).

f[x_,y_]:=((x*y)/(x^2+y^2))(* the function *)

(* First, several basic plots of the function showing the raggedness when x=0 and y=0. *)
Plot3D[f[x,y],{x,-1,1},{y,-1,1},Mesh->None,AxesLabel->{"x","y","z"}]

(* A plot of the level curves, letting Mathematica pick the contours to draw *)

ContourPlot[f[x,y],{x,-1,1},{y,-1,1},ContourLabels->True]

(* A slider showing how the limit as (x,y) approaches (0,0) along different lines (given by y=mx) varies *)

Manipulate[Show[Plot3D[f[x,y],{x,-1,1},{y,-1,1},Mesh->None,AxesLabel->{"x","y","z"}],
ParametricPlot3D[{t,m*t,f[t,m*t]},{t,-1,1},PlotStyle->{Green}]],
{m,-4,4}]

Example III: A slider plot of a function which has a limit as (x,y) goes to (0,0) that shows that the limits along all paths (or at least those with slopes between -4 and 4) thru (0,0) are the same.

g[x_,y_]:= ArcSin[x^2+y^2]
Manipulate[Show[Plot3D[g[x,y],{x,-1,1},{y,-1,1},Mesh->None,AxesLabel->{"x","y","z"}],
ParametricPlot3D[{t,m*t,g[t,m*t]},{t,-1,1},PlotStyle->{Green}]],
{m,-4,4}]

ContourPlot[g[x,y],{x,-1,1},{y,-1,1},ContourLabels->True]