Analysis of Limits using Contour Maps

The following example commands illustrate how you can draw a contour map of a function of two variables that enables you to see that the limit does not exist as (x,y) approaches (0,0). The zhue option shows us in the following contour map that when you approach the origin along straight lines, you can come in on different levels indicated by the different colors. Thus to prove that the limit does not exist try different paths of the form y=kx.

> plot3d((x^2+5*x*y+y^2)/(2*x^2+3*y^2),x=-5..5,y=-5..5,axes=boxed,shading=zhue,style=patchcontour, orientation=[-90,0]);

In the next example, note that each color seems to trace a parabola, and one can get close to the origin along different colors (levels). Thus the limit of the function does not exist as you approach the origin, and to prove that the limit does not exist try different paths of the form y=kx^2.

> plot3d((x^2*y)/(x^4+2*y^2),x=-5..5,y=-5..5,axes=boxed,shading=zhue,style=patchcontour, orientation=[-90,0]);

In the next example, each color seems to trace a curve similar to y=x^3. One can approach the origin along different colors (levels), so the limit of the function does not exist as you approach the origin. Try different paths of the form y=kx^3 to prove that the limit does not exist.

> plot3d((2*x^3*y)/(3*x^6+4*y^2),x=-5..5,y=-5..5,axes=boxed,shading=zhue,style=patchcontour,orientation=[-90,0]);

In the following example, note that the region around the origin is all the same color. This indicates that the limit of the function exists as you approach the origin.

> plot3d((2*x*y^2)/(x^2+y^2),x=-5..5,y=-5..5,axes=boxed,shading=zhue,style=patchcontour,orientation=[-90,0]);

Again, the one color around the origin indicates that the limit of the function exists as you approach the origin.

> plot3d(x^2/sqrt(x^2+y^2),x=-5..5,y=-5..5,axes=boxed,shading=zhue,style=patchcontour,orientation=[-90,0]);

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