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.
> 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.

Note that there is a `Limit` item on the `Constructions`
submenu of the context sensitive menu for a Maple expression. Using
this item will produce a `Limit` command. This is different
from the `limit` command described above and is an example of
what is called an **inert** command in Maple. In the case of the
`Limit` command, all it does is display the mathematical
notation for the limit without attempting to evaluate it. The usual
way to get Maple to evaluate an inert command (like `Limit`) is
to use the `value` command. However, it is almost always
simpler to use the `limit` command and that is what you are
encouraged to do for this lab.

The following definition of limit is given on page 66 of the text.

This definition may seem complicated, but it has a nice graphical interpretation. Plot with the -range set to and the -range set to . Try to choose small enough so that the graph of 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 .

Consider this example. Let and . Then any value for smaller than about 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
. This is too large since the graph intersects the lines and . The value for in the
second `plot` command, however, is small enough since the graph of
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. You may find that using the `smartplot` command
is convenient to use, because you can easily edit the plot ranges. On
the other hand, smart plots take longer to appear. The choice is
yours.

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.

- Consider the following limit statement

The rigorous definition of a limit in this case says that if for each there is a corresponding number such that

First, use the`limit`command to find the limit. Then, given find a value of that works in the definition. Try to find the smallest such . Note that Maple uses the word`infinity`for and that`infinity`can be used in the`limit`command and in a range for the`plot`command, e.g.`x=2..infinity`

. - Consider the following limit.

Use Maple to compute this limit. Then explain why this does*not*imply the existence of the following limit.

2000-09-19