Introduction to the Theory of Limits
1
Introduction.
Consider an object moving on the y-axis. As you have been learning:
1. one can use the record of the object’s position at each time t to calculate average velocity for any
time interval;
2. Seventeenth-Century mathematicians figured out how to extract instantaneous velocity information from average velocity information, but they were unable to put their methods on a solid
logical footing;
3. modern mathematics has provided the needed logical foundation through the mathematics of
limits.
The handout titled The Problem of Velocity discussed the first two of these topics in some detail. This
handout focuses on the third one: its purpose is to give you a rough idea1 of how the theory2 of limits
works. It includes:
• a preliminary verbal definition (the same one you have already seen);
• a step-by-step transition from the preliminary definition to the standard (very technical) definition;
• one example showing how the standard definition is applied;
• some comments on the shape—and the point—of the overall theory.
Throughout, “c” and “L” will stand for particular numbers, and “h” will denote a function that is
defined everywhere in some open interval that contains c, except perhaps at the point c itself.
2
The Preliminary Definition.
One does not define the word “limit” by itself; rather, one defines the entire sentence
“As x approaches c, the limit of h(x) equals L,”
which is written symbolically:
“ lim h(x) = L.”
x→c
1
Limits are quite difficult, and a careful and complete study of them would be too hard and take too much time for
this course.
2
In science and mathematics, the term theory refers to a body of logically-related results (not to a guess about something).
1
Here is the verbal definition:
Verbal Definition
“lim h(x) = L” means: h(x) is guaranteed to be within any prex→c
scribed positive distance of L, provided:
(a), that x is sufficiently close to c; and
(b), that x is not equal to c.
The reason for insisting that x 6= c is that, in most of the important cases, c will not even be in the
domain of h, so that h(c) will not even exist. In order to make the definition work in these cases, it is
important to prevent any potential bad behavior right at the point x = c from interfering; this is done
by completely excluding the point x = c from consideration.3
The intent of this definition is to frame a challenge/response scenario: first, you are challenged with
a maximum positive distance between h(x) and L; then, if you can, you must respond by saying how
close x must be to c to guarantee that h(x) will be this close to L. The statement “lim h(x) = L” is
x→c
the assertion that you will be able to respond to every one of the possible challenges.
In this form, the definition is far too vague: it does not make explicit how to find out whether or not
you will be able to respond to every challenge. The first order of business must therefore be to replace
the verbal definition with a precise equivalent.
3
Transition to the Standard Definition.
I will break the transition into several steps; at each step, I will introduce some revisions in the definition
and explain why the changes leave the meaning intact.
Step 1. I will employ the fact that the distance on the number line between numbers a and b is given
by |a − b|.
Definition, Form #2
“lim h(x) = L” means: |h(x) − L| is guaranteed to be smaller than
x→c
any prescribed positive number, provided:
(a), that |x − c| is sufficiently small; and
(b), that x is not equal to c.
Step 2. I will introduce a symbol for the “prescribed positive number;” for some reason, the Greek
lowercase epsilon () is always used for this. Working into the definition also enables me to replace
“smaller than” by “<”.4 Here is the result.
3
There will be many cases, however, in which h(c) exists and is equal to L. When this happens, one says that h is
continuous at the point x = c.
4
I will also replace the word “any” with the more precise “every.”
2
Definition, Form #3
“lim h(x) = L” means: For every number > 0, it is guaranteed
x→c
that |h(x) − L| < , provided:
(a), that |x − c| is sufficiently small; and
(b), that x is not equal to c.
Step 3. Note that the the restriction “sufficiently small” means “less than some sufficiently small
(positive) number.” I will introduce a symbol for this number; for some reason, Greek lowercase delta
(δ) is always used here. Working δ into the definition allows me to replace “is sufficiently small” with
“< δ” and makes it a simple matter to combine conditions (a) and (b). The result is:
Definition, Form #4
“lim h(x) = L” means: For every number > 0, it is guaranteed
x→c
that |h(x) − L| < , provided that
0 < |x − c| < δ
for some sufficiently small positive number δ.
Step 4. I will rearrange the order to emphasize the challenge/response structure: you are challenged
with the number , and you must respond with the number δ.
Definition, Form #5
“lim h(x) = L” means: For every [number] > 0, there exists a [numx→c
ber] δ > 0 with the property: it is guaranteed that |h(x) − L| < ,
provided that
0 < |x − c| < δ.
Step 5. Form #5 is quite close to the standard definition. In this final step, I need only introduces
som streamlining in order to arrive at the standard definition.
Definition, Form #6 (Standard Form)
“lim h(x) = L” means: For every > 0, there exists a δ > 0 such
x→c
that: whenever 0 < |x − c| < δ, then |h(x) − L| < .
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4
An Example.
Let me return to the flawed velocity computation in the handout on p.5 of the handout titled The
Problem of Velocity.: I will prove, using the standard definition (form #6) that
lim
∆t→0
f (a + ∆t) − f (a)
= −32a.
∆t
(1)
The first step, which I will not repeat here,5 is to show algebraically that for all ∆t 6= 0,
f (a + ∆t) − f (a)
= −32a − 16∆t;
∆t
(2)
because (1) requires ∆t 6= 0, (2) enables me to replace assertion (1) with the equivalent simpler assertion
lim ( − 32a − 16∆t) = −32a.
(3)
∆t→0
Here is the proof of (3). Suppose > 0 has been chosen.6 For this , we must find a δ > 0 (a response)
such that
0 < |∆t| < δ =⇒ ( − 32a − 16∆t) − ( − 32a) < or (after simplifying) such that
0 < |∆t| < δ =⇒ |(−16)∆t| < (4)
Now, observe that (−16)∆t = 16|∆t|, so that
(−16)∆t < ⇐⇒
16|∆t| < ⇐⇒ |∆t| < ;
16
(5)
observe further that, obviously,
0 < |∆t| <
16
=⇒ |∆t| <
.
16
This says that, whatever the (challenge) value of may be,
you may take δ equal to
as your response: this value of δ will make (4) true.
16
This proves that the conditions of the definition have been satisfied: for every > 0 there is a δ > 0
that makes (4) true.
5
6
This was done in the Problem of Velocity handout.
that is: that the challenge has been issued.
4
5
Some Comments.
In this final section, I want to convey a sense of where the theory goes from here and say more about
the point of the entire enterprise.
5.1
Where the Theory Goes from Here.
Limit Laws. The first order of business is prove certain general properties of limits; for example, one
of them states,
If lim f (x) and lim g(x) both exist, then
x→c
x→c
lim f (x) + g(x) = lim f (x) + lim g(x) .
x→c
x→c
x→c
(6)
This statement is obvious to the intuition; after all, for example, if f (x) is very close to 5 and g(x) is
very close to 3, then (f (x) + g(x)) is going to be very close to 8. Most of the other limit laws are just
as obvious, but many of them are nevertheless tricky to prove. It is natural to ask: Why would anyone
bother with such “proofs of the obvious”? There are several reasons.
1. The successful proof of a limit law increases confidence that the complicated definition really does
capture the intuitive idea of a limit. If the definition did not make it possible to prove statements
like (6), mathematicians would scrap the definition and start over.
2. Once the limit laws have been proved, they are on rock-solid logical ground. They can therefore
be used with complete confidence to compute particular limits. (And they are used in these ways,
because they often bypass –δ proofs in situations where such proofs would be mind-bogglingly
difficult.)
3. The limit laws frequently enable you to discover the value of a limit. The same cannot be said of
the –δ definition: if you wish to use the –δ definition to prove that that lim f (x) = L, you must
x→a
already know the value of L before you can even begin.
Beyond Limit Laws. As soon the limit laws and other such results are established, they can be enlisted
as research tools to advance the theory even further, with the upshot that it now extends far beyond the
bounds of naı̈ve intuition7 and is capable of revealing surprising facts. For example, it turns out that
there exist continuous curves that fail to have even a single tangent line! This highly countintuitive fact
shocked the mathematical community when it appeared, but mathematicians adjusted their intuitions
to accommodate this new knowledge, because the proof is unarguably correct.
5.2
The Point of the Theory.
We have already discussed the original justification for the theory: it provides a firm logical underpinning
for the concept of instantaneous velocity. It thus completes our response to Zeno’s paradox and achieves
what Zeno thought impossible: it links precise reasoning to the notion of limits (and therefore to the
phenomena of motion). To my mind, this link is the best argument for the importance of the theory; it
is only by harnessing logic that limit theory has been able to expand beyond our intuitions and show
us things (like tangentless curves) that we never would have expected.
7
By “naı̈ve intuition,” I mean an intuition—gut feeling—that has not been refined or changed by something like a
surprising mathematical result.
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