MATH 163, Spring 2001 Due Date: Name(s):
Project 2.2: Guessing Limits Numerically
Objective
To guess the limit of a function at a point numerically.
Narrative
If you have not already done so, read Section 2.2 in the text.
Prior to having theorems on limits at our disposal, there are two major issues surrounding the limit of a
function at a point: The first is guessing what the limit is, if it even exists; this issue can often be approached
either graphically or numerically. The second issue involves proving that the guess you made is correct; this
issue involves using the formal definition of limit.
In this project we address the issue of guessing limits numerically. In Project 2.4 we address the issue of
proving a guess is correct. In this project we also illustrate how to perform repeated computations efficiently
in Maple using a "do loop".
Task
a) Type the command lines in the left-hand column below into Maple in the order in which they are listed.
These commands will help you estimate limx0
1 - cos x
x2
numerically, if it exists. (Note: It's OK to type
the entire first loop for n from 1 to 6 do ... od: on one line, and the entire second loop for n from
1 to 6 do ... od: also on one line.)
> # Project 2.2: Guessing Limits Numerically
> restart; Clear Maple's memory.
> f := x -> (1-cos(x))/x^2; Let f(x) = (1 - cos x)/x2
.
> plot(f(x),x=-1..1); Plot the graph of f.
> a := 0.0; We are interested in estimating limxa f(x).
> f(a); What is f(a)?
> for n from 1 to 6 Let's look at the values of f(x) for x < a.
do This is the beginning of a "do loop".
x := a-1/2^n: Let x = a - 1
2n .
print(evalf(x), evalf(f(x))); Print the values of x and f(x).
od: This is the end of our "do loop".
> for n from 1 to 6 Now let's look at some values of f(x) for x > a.
do
x := a+1/2^n:
print(evalf(x), evalf(f(x)));
od:
b) On the basis of this data, do you think limx0
1 - cos x
x2
exists? If so, what do you think it is (to 4
decimal places of accuracy)? Justify your answer.
Your lab report will be a hard copy of your typed input and Maple's responses (including both text and
graphics), together with your written response.
Comments
1. You can guess whether or not limx0
1 - cos x
x2
exists, and if it does exist, what its value is, on the basis
of numerical "evidence" (as we did in this project), but you cannot say for sure that you're correct: you
can never perform more than a finite number of computations, and however close x is to 0, you may
miss some critical behavior of f(x) =
1 - cos x
x2
that might affect your guess. It is because of this that
we must turn to the formal concept of the limit.
2. Different rates of convergence can be achieved by replacing 1/2^n by 1/n^2 (this produces a slower rate
of convergence) or 1/n^n (this produces a faster rate of convergence).
3. The physical limitations of your computer may limit the accuracy of your computations.
4. Maple has a built in command limit(f(x),x=a) that allows you to compute (some) limits automatically.
(Variations on this command include limit(f(x),x=infinity) and limit(f(x),x=-infinity) for
computing limits at , and limit(f(x),x=a,left) and limit(f(x),x=a,right) for computing
left- and right-hand limits.) Since we are interested not just in what limits are, but how they are
computed, we intentionally avoided using this command in this project.
5. At the end of the do loops in the above code, Maple will think that n = 6 and x = a  1/2n
. (You can
check this by entering the commands n; and x; after each loop.) This is important to know since if,
subsequent to the appropriate do loop, you wanted to reuse n or x as a variable then you would have
to redefine it as a variable using the command n = ´n´ or the command x = ´x´.