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The purpose of this lab is to use Maple to become more familiar with
limits of functions, including onesided 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 undefined
or sometimes 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 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.
Definition 1
We say that the number
is the
limit of
as
approaches
provided that, given any number
, there exists a number
such that
for all
such that
This definition may seem
complicated, but its graphical interpretation is not so bad. It says
that if you plot with the range set to
you can always choose a value of
small enough so that when you shrink the plot range to
and plot the function, its graph will not
intersect the top or the bottom edges of your plot. For example,
suppose , 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);
>
L := 4; epsilon := 0.2; delta := 0.1;
>
plot({epsilon, epsilon, f(x)  L},x=2delta..2+delta);
>
delta := 0.048;
>
plot({epsilon, epsilon, f(x)  L},x=2delta..2+delta);
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. Note that by
using labels for the values of , , and , we
were able to use the same plot command. By copying and pasting, you can
save yourself some typing. Using labels also helps prevent errors.
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 righthand and lefthand limits. Loosely speaking, the
righthand limit of at is if approaches as
approaches from the right. That is, the values of satisfy
. The lefthand limit is defined in an analogous manner, with
the values of approaching 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 want to define your own
piecewisedefined function, then the Maple piecewise command
is the best way to do it. Suppose you wanted to define the following
function.
Then the Maple command would be the following.
>
g := x > piecewise(x < 0, x, x^2+1);
If you want to see your function in a more familiar form, just run a
command like the one below.
>
g(x);
The way the piecewise command works is that you give it a
sequence of pairs of conditions and formulas that define your
function. When you want to evaluate your function at a particular
value of , Maple checks the conditions from left to right until it
finds the one that your value of satisifies. It then plugs the
value of into the next formula. However, notice that the command
above only has one condition and two formulas. This is because any
value of is either less than zero or it is greater than or equal
to zero, so if a particular value of fails the first condition,
i.e. is not less than zero, it must be greater than or equal to zero
and the second formula is the one to use. For more information, see
the help page for piecewise.
The limit command works fine for functions that are defined
via the piecewise command, as shown in the example below.
>
limit(g(x), x=0);
>
limit(g(x),x=0, left);
>
limit(g(x),x=0, right);
>
plot(g(x), x=0.1..0.1);
 For the functions and values of 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 , close
to the maximum such value of delta, that
works for
.

.


 Find the right and lefthand limits of the following function at
. Also, plot the function and
relate your limits to the graph.
Does
exist? Explain your reasoning.
 consider the following limit statement.
If
, find an appropriate . Try to repeat your
procedure with
. What conclusion do you draw from
your work? Explain.
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Dina Solitro
20010116