Limits of Functions

You can take the limit of a function as the variable approaches 
a fixed number for a defined function f:
> f := x -> (x^2 -4)/(x-2);

                                      2
                                     x  - 4
                           f := x -> ------
                                     x - 2

> limit(f(x), x=2);

                                  4

Graph the function to check if this limit seems correct:
> plot(f, -5..5);

> factor(f(x));

                                x + 2

You can find limits of more complicated functions:
> f := x -> (x-4)/(sqrt(x)-2);

                                     x - 4
                        f := x -> -----------
                                  sqrt(x) - 2

> limit(f(x),x=4);

                                  4

Graph the function to check the answer:
> plot(f, 0..5);

You can determine limits at infity as well:
> g := x-> sqrt(x^2- 4*x) -x;

                                     2
                     g := x -> sqrt(x  - 4 x) - x

> limit(g(x), x=infinity);

                                  -2

> plot(g, 0..100);

> plot(g, 100..1000);

You can examine piecewise defined functions at the points where their rules change.
At these points, you use Maple's ability to determine left and right limits.
> f := proc(x)
> if x < 0 then x-1
>  else x^2 fi
> end;

           f := proc(x) if x < 0 then x - 1 else x^2 fi end

> plot(f, -2..2);

You can use one-sided limits:
> limit(x-1, x=0, left);

                                  -1

> limit(x^2, x=0, right);

                                  0

>