Laboratory I.3
Introduction to Limits of Functions
Goals
• The student will develop an intuitive understanding of the concepts of the limit.
• The student will be able to determine the existence of the limit of a function by using tables to
look at the numerical values of the function, by looking at the graph of the function, and by using
the
Calculus:Limit
command.
• The student will explore the solution to a problem by applying the notion of the limit.
Before the Lab
In this lab, we'll be looking at how a function behaves as
x
values approach, but do not
reach, a certain "target" value. Here's a really simple example with the function
f
(
x
) =
x
2
.
x
f(x)
2.5
6.25
2.9
8.41
2.99
8.9401
2.999
8.994001
What is the "target" value that the
x
values are getting closer and closer to? What is the resulting
pattern in the values of
f
(
x
)?
These are the typical questions that we will be working with in this
lab.
We'll spend the rest of the time in this "Before the Lab" section talking about efficient ways of
creating these tables in DERIVE, and leave the Lab time itself for working on the mathematics.
DERIVE uses square brackets [ ] to designate a list of things which should be kept in the
order they are typed. The resulting mathematical object is called a vector. Thus, if you wanted
the first pair of (
x
,
y
) values above, you would
Author
[2.5, 6.25]
because you need the numbers to appear in that order (that's why we call it an ordered pair). To
recreate the whole table, we need a vector of ordered pairs, so we would
Author
[[2.5,6.25], [2.9,8.41], [2.99,8.9401], [2.999,8.994001]]
You can
Plot
the result in DERIVE and see not a curve but just the four points.
Now, you'll get pretty tired typing all the numbers in, plus having to figure out all the
y
values
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MATH 2413
Lab 3
Page 2 of 4
first, so here's a more efficient way to create a table like the one above. Suppose we wanted a
table of five points, at distances 1/2, 1/4, 1/8, 1/16, and 1/32 below
x
= 3. DERIVE can make any
table like this in one command, as long as the
x
values follow some pattern (in the example
above, the first point is not in the same pattern as the later three). The pattern in the
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 Spring '10
 Moody
 Limit, lim, Binary relation

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