Laboratory 4 
Limits and Continuity of Multivariable Functions
Name:

Code

basicContourPlot[f_,{x,x0_,x1_},{y,y0_,y1_},{z,z0_,z1_}]:=Module[{a,b,c,dz},dz=(z1-z0)/20;
a=Plot3D[f[x,y],{x,x0,x1},{y,y0,y1},
PlotRange->{z0,z1},MeshFunctions->{#3&},AxesLabel->{"x","y","z"}];
Do[c[i]=ContourPlot[f[x,y]==z0+i dz,{x,x0,x1},{y,y0,y1},
ColorFunction->Function[{x,y},RGBColor[i/20,0,1-i/20]],Frame->True];,{i,0,20}];
b=Show[Table[c[j],{j,0,20}],PlotRange->{{x0,x1},{y0,y1}}];
Rasterize[Show[GraphicsRow[{a,b}],ImageSize->550]]]

basicContourPlot3D[f_,g_:1,{x,x0_,x1_},{y,y0_,y1_},{z,z0_,z1_},{w,w0_,w1_},pt_:0]:=
Module[{c,u,v,p,p1,p2},
u[x_,y_,z_]=f[x,y,z];
v[x_,y_,z_]=If[g==1,1,{},g[x,y,z]];
p=If[pt==0,{x0-1,y0-1,z0-1},pt];
p2=Graphics3D[{PointSize[0.02],Point[p]}];
Manipulate[
p1=ContourPlot3D[u[x,y,z]==c*v[x,y,z],{x,x0,x1},{y,y0,y1},{z,z0,z1},
Axes->True,Mesh->None,BoxRatios->Automatic,ContourStyle->Opacity[0.9],PlotLabel->u[x,y,z]/v[x,y,z]==c,
PlotRange->{{x0,x1},{y0,y1},{z0,z1}}];
Rasterize[Show[p1,p2]],
{{c,(w0+w1)/2,"level"},w0,w1}]
]

Exercises

Exercise 1

For each of the following functions, use graphical methods to estimate the value of (lim_((x,y)->(a,b)))f(x,y), if possible. If f(a,b) is undefined, is it possible to assign a value to f(a,b) to make f continuous at (a,b)?	
(a) f(x,y)=((x+y-4(x^2)(y^2))/((x^2)+(y^2)-5)); a=1,b=1		(b) f(x,y)=((x+y-4(x^2)(y^2))/((x^2)+(y^2)-5)); a=2,b=1
(c) f(x,y)=(((1+xy))^(1/xy)); a=0, b=0

Exercise 2

For each of the following functions plot three dimensional graphs and contour graphs (use basicContourPlot to generate them) to determine if the limit (lim_((x,y)->(0,0)))f(x,y) exists. 
(a) f(x,y)=((x y)/(Sqrt[(x^2)+(y^2)]))		(b) f(x,y)=(((y -x))/(Sqrt[(x^2)+(y^2)]))
(c) f(x,y)=(((x^3)+(y^3))/((x^2)+(y^2)))

Exercise 3

For each of the following functions use level surfaces (use basicContourPlot3D to generate them) to determine if the limit (lim_((x,y,z)->(0,0,0)))f(x,y,z) exists.
(a) f(x,y,z)=(xyz/((x^2)+(y^2)+(z^2)))		(b) f(x,y,z)=((z^2)/(x y+(z^3)))

Exercise 4

Use level surfaces to determine if (lim_((x,y,z)->(0,0,0)))((cos(2x+y))/(x y+(z^3))) exists. How would you describe the change in the behavior of the level surfaces f(x,y,z)=c as c gets large? What does that say about the limit of f(x,y,z) as (x,y,z)->(0,0,0)?