The purpose of this lab is to use Maple to become more familiar with limits of functions, including one-sided limits.

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.

It should be no secret by now that for most
functions defined by a single formula, when exists. For more complicated functions, this
may not be true. For dealing with some of these exceptional cases, we
need to define right-hand and left-hand limits. Loosely speaking, the
right-hand limit of at **a** is **L** if approaches **L** as
**x** approaches **a** from the right. That is, the values of **x** satisfy
**x > a**. The left-hand limit is defined in an analogous manner, with
the values of **x** approaching **a** from the left. Maple can
compute these special limits with commands like those shown below. The
Maple `floor` function is actually the greatest integer function.

> plot(floor(x),x=0..4);

> limit(floor(x),x=1,right);

> limit(floor(x),x=1,left);

The `floor` function is one of Maple's defined functions, so you
might expect things to work properly. If you define your own function
in a piecewise fashion, however, things don't go so smoothly. See the
example below.

> f := x -> if x < 0 then -x else x^2+1 fi ;

f := proc(x) options operator,arrow; if x < 0 then -x else x^2+1 fi end

> limit(f(x),x=0);

Error, (in f) cannot evaluate boolean

> limit(f(x),x=0,left);

Error, (in f) cannot evaluate boolean

> limit(f(x),x=1);

Error, (in f) cannot evaluate boolean

The problem is that Maple can't evaluate the if statement in the
definition of unless **x** has a numerical value. The solution in
a case like this is to plot the graph of the function or compute
values of the function near **a**, as shown in the following
examples. The quotes in the `plot` command in the example tell
Maple to wait until it has a numerical value for **x** before it tries
to evaluate .

> plot('f(x)',x=-0.1..0.1);

> evalf(f(-0.01));

> evalf(f(-0.0001));

Using these commands as examples, you should be able to figure out that and that .

- For the functions and values of
**a**given below, go through the following steps.

- i.
- Find whether the exists or not. If it does, determine the limit.
- ii.
- For the limits that exist, find a value of that
works for .

- .

- Find the right- and left-hand limits of the following function at
**x=1**. Also, plot the function and relate your limits to the graph.Does exist? Explain your reasoning.

- Suppose that the function is defined by
where

**a**and**b**are parameters. Can you find values for**a**and**b**that will make continuous at**x=0**? Justify your answer.

Mon Jun 26 13:37:30 EDT 1995