MATH 261, Spring 2001 Due Date: Name(s):
Extra Project 15.2a: Limits and Continuity
Objective
If you have not already done so, read Section 15.2 of the text. In this project we investigate why limits of
functions of two variables may fail to exist.
Narrative
It may not be easy to see why lim(x,y)(a,b) f(x, y) fails to exist even though f(x, y) exists for every value
of (x, y) close to (a, b), but in some cases Maple can help.
Task
1. Type the command lines below into Maple in the order in which they are listed. They draw the graph
of a function f(x, y) for which lim(x,y)(0,0) f(x, y) fails to exist -- since limx0 f(x, 0) = limy0 f(0, y)
-- even though f(x, y) exists for every value of (x, y) close to (0, 0).
> # Project 15.2a: Limits and Continuity
> restart: with(plots);
> # Task 1
> f := (x,y) -> (2*x^2+y^2)/(x^2+y^2);
> plot3d(f(x,y),x=-10..10,y=-10..10);
2. Continue by typing the command lines below into Maple in the order in which they are listed. They
draw the graph of a function g(x, y) for which lim(x,y)(0,0) g(x, y) = 0 when approaching the origin
along any line that passes through the origin, but lim(x,y)(0,0) g(x, y) fails to exist since, if we approach
the origin along the curve x = t3
, y = t, we find that lim(x,y)(0,0) g(x, y) = 0.5.
> # Task 2
> g := (x,y) -> x*y^3/(x^2+y^6);
> plot3d(g(x,y),x=-10..10,y=-10..10);
At this time, make a hard-copy of your typed input and Maple's responses. Then, ...
3. On the basis of the graphic you drew in Task 1, explain (in words) why, lim(x,y)(0,0) f(x, y) fails to
exist even though f(x, y) exists for every value of (x, y) close to (0, 0).
4. On the basis of the graphic you drew in Task 2, explain (in words) why, lim(x,y)(0,0) g(x, y) fails to
exist even though g(x, y) exists for every value of (x, y) close to (0, 0).
(You may need to move around the graphs you created in Tasks 1 and 2 to do Tasks 3 and 4. You may
also want to zoom-in on the origin by changing the ranges of x and y.)