Limits and Continuity

by John Hebron, SFU, October 1999

Example 1: Different limit along x=0 and y=0.

> f1:=(x^2-y^2)/(x^2+y^2);

The 3-D plot of f1

Example 2: Different limit along x=0 and y=x.

> f2:=x*y/(x^2+y^2);

The 3-D plot of f2

Example 3: Different limit along y=mx and x=y^2.

> f3:=x*y^2/(x^2+y^4);

The 3-D plot of f3

Example 4: A function continuous on r>1 in cylindrical coordinates.

Note that to plot this function in cylindrical coordinates, it must be specified in parametric form as [r, theta, z(r,theta)]. This is equivalent to example 10, section 12.2, page 756 of the textbook.

> f4:=[r,theta,ln(r^2-1)];

The plot of f4 in cylindrical coordinates