Section 14.2 Limits and Continuity

The limit of f(x,y) at (0,0). 3d plot of a function associated with Table 1. 

f(x,y) := sin(x^2+y^2)/(x^2+y^2);

f(0,0);

draw3d(enhanced3d=true, view=[30,135],
       explicit(f(x,y),x,-1,1,y,-1,1),
       xlabel="x", ylabel="y", surface_hide=true,
       user_preamble="set size 1,1.25");

Level Curves: contour_levels=[zmin,increment,zmax] determines the level curves of heights from zmin to zmax by
value of increment, but some lines may not appear in the graph.

wxdraw3d(color=red, contour=map, contour_levels=[0.5,0.02,1.0],
       explicit(f(x,y),x,-1,1,y,-1,1),
       user_preamble="unset clabel; set size 0.8,1.2", xlabel="x", ylabel="y");

Example 1 & 7. The limit of f(x,y) at (0,0). 3d plot of a function associated with Table 2.

f(x,y) := (x^2-y^2)/(x^2+y^2);

f(0,0);

draw3d(enhanced3d=true, view=[30,135],
       explicit(f(x,y),x,-1,1,y,-1,1),
       xlabel="x", ylabel="y", surface_hide=true,
       user_preamble="set size 1,1.25");

Level Curves: contour_levels=[zmin,increment,zmax] determines the level curves of heights from zmin to zmax by value of increment, but some lines may not appear in the graph

wxdraw3d(color=red, contour=map, contour_levels=[-0.9,0.2,0.9],
       explicit(f(x,y),x,-1,1,y,-1,1),
       user_preamble="unset clabel; set size 0.8,1.2", xlabel="x", ylabel="y");

Example 1. Show the limit at (0,0) does not exist. limit(f(x),x,a) computes the limit of f(x) as x approaches a.

limit(f(x,0),x,0);

limit(f(0,y),y,0);

limit(f(t,t),t,0);

limit(f(t,2*t),t,0);

Example 4. Limits and continuity of polynomial functions.

f(x,y) := x^2*y^3 - x^3*y^2 + 3*x + 2*y;

f(1,2);

draw3d(enhanced3d=true, view=[30,135],
       explicit(f(x,y),x,-2,4,y,-1,5),
       xlabel="x", ylabel="y", surface_hide=true,
       user_preamble="set size 1,1.25");