> # LIMITS of FUNCTIONS 
> #
> # We will want to use plot3d.
> with(plots):
> #
> # FIRST FUNCTION:
> #
> # This function does not have a limit at the origin. The limit of f(x,y)
> # as we move towards
> # the origin depends on the direction we use.
> #
> 
> f:=x*y/(x^2+y^2);

                                    x y
                             f := -------
                                   2    2
                                  x  + y

> plot3d(f,x=-2..2,y=-2..2);

> # The function has a ridge along the line x=y. Every point on this line
> # has value 0.5, except
> # for the origin. If we look at the contourplot, we see that the
> # contours are very close to one
> # another near the origin.
> contourplot(f,x=-2..2,y=-2..2);

> #
> # SECOND FUNCTION:
> #
> #  This function has the limit of zero as we approach the origin from
> # any direction.
> # However, the function has a different limit if we look at the parabola
> #  y=x^2.
> # Again, the contours are tightly bunched near the origin.
> g:=x^2*y/(x^4+y^2);

                                    2
                                   x  y
                             g := -------
                                   4    2
                                  x  + y

> plot3d(g,x=-2..2,y=-2..2);

> contourplot(g,x=-2..2,y=-2..2);

> #
> # THIRD FUNCTION:
> #
> # This function does have a limit as (x,y) tends to (0,0).
> # Notice that the contours don't show the same bunching as in the
> # previous two examples.
> h:=2*x^2*y/(x^2+y^2);

                                     2
                                    x  y
                            h := 2 -------
                                    2    2
                                   x  + y

> plot3d(h,x=-2..2,y=-2..2);

> contourplot(h,x=-2..2,y=-2..2);

> #
> # Now look at a COMPOSITION of two continuous functions.
> # x^2+y^2 is continuous everywhere. tan(r) is continuous if Pi/2 < r <
> # 3Pi/2
> p:=tan(sqrt(x1^2+y1^2));

                                      2     2
                      p := tan(sqrt(x1  + y1 ))

> # We need to use polar coordinates to plot this, because we want to
> # control the value of x^2+y^2
> x1:=r*cos(t); y1:=r*sin(t);

                            x1 := r cos(t)


                            y1 := r sin(t)

> plot3d([x1,y1,p],r=Pi/2+0.1..3*Pi/2-0.1,t=0..2*Pi);

>