Limits of many functions and expressions can be computed in Maple with
the `limit` command. Some examples are given below.

> limit(x^2+2*x,x=2);

> limit(sin(x)/x,x=0);

> f := x -> (x+3)/(x^2+7*x+12) ;

> limit(f(x),x=-3);

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

If the limit exists, Maple can usually
find it. In cases where the limit doesn't exist, Maple gives the
answer `infinity`

for an unbounded limit or gives a range like
`-1..1`

if the limit doesn't exist, but the expression or
function is bounded. See the examples below.

> limit(1/x,x=0);

> limit(sin(1/x),x=0);

You can also use Maple to compute limits as **x** goes to
as shown below.

> f(x);

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

> limit(f(x),x= -infinity);

The formal definition for a limit is given below.

This definition may seem
complicated, but its graphical interpretation is not so bad. It says
that if you plot with the **y** range set to
you can always choose a value of
small enough so that when you shrink the **x** plot range to
and plot the function, its graph will not
intersect the top or the bottom edges of your plot. For example,
suppose , **a=2** and . Then any value of
smaller than about will work. To see what is going
on, look at the plots generated by the following commands.

> f := x -> x^2;

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

> plot({-0.2,0.2,f(x)-4},x=2-0.1..2+0.1,y=-0.2..0.2);

> plot({-0.2,0.2,f(x)-4},x=2-0.048..2+0.048,y=-0.2..0.2);

In the first of the two `plot` commands, the value of is
. This is too large, since the graph intersects the lines
and . The value of for in the second
`plot` command, however, is small enough, since the graph of
goes off the sides of the plot. Make sure that you understand
this example. If you don't understand, ask for help.

Thu Sep 7 15:27:05 EDT 1995