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Limits of many functions and expressions can be computed in Maple with
the `limit` command.

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

> g := x -> (3*x^2+4)/(x^2+7*x+12) ;

> limit(g(x),x=-2);

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

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

`-1..1`

if the limit doesn't exist, but the expression or
function oscillates within these bounds.
> limit(1/x,x=0);

> limit(1/x^2,x=0);

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

Maple can also do one-sided limits. Use Maple's online help to find out how to handle such limits.

The following definition of limit is given on page 72 of the text.

This definition may seem complicated, but it has a nice graphical
interpretation. Plot *f* with the *y*-range set to and the *x*-range set to . Try
to choose small enough so that the graph of *f* stays
between the top and bottom of the plot. If you can find such a
no matter how small the , then the limit exists
and is equal to *L*.

Consider this example. Let and . Then any value for smaller than about 0.049 will work. To see why this is so look at the plots generated by the following commands.

> f:=x->x^2;

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

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

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

In the first of the two `plot` commands, the value of is
0.1. This is too large since the graph intersects the lines *y* = 4
- 0.2 and *y* = 4 + 0.2. The 0.048 value for in the
second `plot` command, however, is small enough since the graph of
*f* goes out the sides of the plot. Make sure you understand what is
done in this example since you will need to do similar work in some of
the exercises.

In Exercises 1 and 2, use the `limit` command to find
. Then experiment with some
plots to find a that works for Also, find
a that doesn't work. These two values should be
close to each other. Thus your that works should be near
the largest value of that can be used. Although, you will
probably need to do many plots, submit only the two required plots
with your worksheet. These plots should be in the style of the
example above. Next, repeat the problem with .Conclude with a one paragraph statement of what you have learned in
the exercise.

- 1.
- 2.
- 3.
- Consider the following limit statement

If , find an appropriate . Repeat your procedure with . What conclusion do you draw from your work? Why?

9/14/1998