Chapter 22 Finite Limits as The notion of infinity has long been troublesome both philosophically and mathematically The ancient Greeks essentially banished infinity from mathematics and in doing so may have delayed the discovery of calculus for more than 2000 years ID: 710496
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Slide1
Limits Involving Infinity
Chapter 2.2Slide2
Finite Limits as
The notion of infinity has long been troublesome both philosophically and mathematically
The ancient Greeks essentially banished infinity from mathematics and in doing so, may have delayed the discovery of calculus for more than 2,000 years
An example known as Zeno’s Paradox, attributed to the Greek philosopher Zeno of Elea, will illustrate why
2Slide3
Finite Limits as
Suppose that your are walking in a straight line from your seat to the door
Before you get there, you must walk exactly half the distance
Before you get to the half-way point, you must walk half of that distanceBefore you get to the quarter-distance, you must walk half of that distance, and so on…
Therefore, you will never get to the door
But, of course, you do get to the door so we have a paradox
3Slide4
Finite Limits as
The problem, in Zeno’s view, was that you would have to perform an infinite number of tasks, which he maintained is impossible
There are several other similar paradoxes, all of which are meant to show why infinity makes no sense. (If you want to read more, Google “Zeno’s Paradoxes”)
The point for us to remember is that we must be careful when working with infinity; we must be precise in what we mean by this term
4Slide5
Finite Limits as
The first thing to remember is that infinity is not a number, rather it is a quality of endlessness
We will use the notation
and
, and these will mean that we allow
to grow “as big as we like”
Note the similarity to the phrase “as close as we like” or synonymously, “as small as we want”
Working with
infinite limits
or with limits
at infinity
requires definitions that are similar to the limit definition (though we won’t use them in this class)
5Slide6
Finite Limits as
Consider the function
What happens to the function values as
?
That is, what do the function values tend to become if we allow
to become larger and larger?
If
, then
; if
, then
; if
, then
; if
, then
What number is
approaching?
6Slide7
Finite Limits as
7Slide8
Finite Limits as
What happens with
values as
?
That is, what do the function values tend to become if we allow
to continue to the left as far as we like?
If
, then
; if
, then
; if
, then
; if
, then
What number is
approaching?
8Slide9
Finite Limits as
9Slide10
Finite Limits as
The graphs demonstrate the following (these are basic so you MUST know them):
These two facts are
theorems
that we can prove from a precise definition; we will take them as given
Note that the line
is a horizontal asymptote
10Slide11
Horizontal Asymptote
DEFINITION:
The line
is a
horizontal asymptote of the graph of a function
if either
11Slide12
Properties of Limits as
THEOREM:
If
,
, and
are real numbers and if
Sum/Difference Rule:
Product Rule:
Constant Multiple Rule:
12Slide13
Properties of Limits as
THEOREM:
If
,
, and
are real numbers and if
Quotient Rule:
Power Rule: if
and
are integers,
, then
provided that
is a real number.
13Slide14
Properties of Limits as
These properties are analogous to the general properties of limits from the previous section
Again note that these are theorems that we can prove
Concerning th
e Power Rule, this means that we can do the following with radicals
14Slide15
Example 1: Looking for Horizontal Asymptotes
Find
and
.
15Slide16
Example 1: Looking for Horizontal Asymptotes
The method demonstrated here is very useful for finding limits at infinity of rational functions. We will divide all terms in the numerator and denominator by the largest power of
, then use the limit properties to evaluate (and the fact that
. This is the same as multiplying by
where
is the largest exponent.
16Slide17
Example 1: Looking for Horizontal Asymptotes
17Slide18
Example 1: Looking for Horizontal Asymptotes
18Slide19
Example 1: Looking for Horizontal Asymptotes
19Slide20
Example 2: Finding a Limit as
Use the Sandwich Theorem to find
20Slide21
Example 2: Finding a Limit as
We know that
If
, then
Now,
By the Sandwich Theorem,
21Slide22
Example 3: Using the Limit Properties for
Find
22Slide23
Example 3: Using the Limit Properties for
23Slide24
Infinite Limits as
When finding limits as
, we may say that we are finding limits “at infinity”
If, on the other hand, we wish to find a limit as
, where
is a real number, and if the function values increase/decrease without bound, then we may say we are finding “infinite limits”
We will write
if the function value increase without bound as
We will write
if the function values decrease without bound as
24Slide25
Infinite Limits as
Ordinarily, to write something like
makes no sense because infinity is not a number
However, if we define what this means (as we did in the previous slide), then we are allowed to use the above notation
Remember, though, that if function values increase/decrease without bound as
, then the limit
does not exist
It is sometimes desirable to know whether the function values continue to increase or continue to decrease without bound
25Slide26
Vertical Asymptotes
DEFINITION:
The line
is a
vertical asymptote of the graph of a function
if either
If
and
, then we may write
Note that in each case above, the limit DNE!
26Slide27
Example 4: Finding Vertical Asymptotes
Find the vertical asymptotes of
. Describe the behavior to the left and right of each vertical asymptote.
27Slide28
Example 4: Finding Vertical Asymptotes
Since
, then by the definition of vertical asymptotes this is sufficient to conclude that
is a vertical asymptote. However, it is also the case that
Therefore, whether approaching zero from the right or the left, the function values increase without bound. We can write
. (Is the same true of
?)
28Slide29
Example 5: Finding Vertical Asymptotes
Find all vertical asymptotes for the graph of
.
29Slide30
Example 5: Finding Vertical Asymptotes
Since
, then the asymptotes are found at all values of
where
. These occur at
; that is, at odd multiples of
. The asymptotes are at all values of
such that
where
is an integer.
For each value of
,
30Slide31
Example 5: Finding Vertical Asymptotes
31Slide32
End Behavior Models
By “end behavior” we mean
For some functions, we can use a simpler “model” of a function that has the same end behavior
That is, both functions approach the same value as
approaches positive or negative infinity
In the case of polynomial functions written in standard form, the lead term can be use to model end behavior
32Slide33
Example 6: Modeling Functions for
Large
Let
and
. Show that, while
and
are quite different for numerically small values of
, they are virtually identical for
large.
33Slide34
Example 6: Modeling Functions for
Large
34Slide35
Example 6: Modeling Functions for
Large
35Slide36
Example 6: Modeling Functions for
Large
If we take the limit of the ratio of the functions as
, we see that the limit approaches 1 (confirming what was seen in the graphs):
36Slide37
End Behavior Model
DEFINITION:
The function
is:a
right end behavior model for if and only if
a
left end behavior model
for
if and only if
If one function provides both a left and right end behavior model, it is simply called an
end behavior model
.
37Slide38
End Behavior Model
To clarify this definition, a function
is a right end behavior model for a function
if both functions have nearly similar values for “large enough” positive values of
A function is a left end behavior model for a function
if both functions have nearly similar values for “large enough” negative values of
Remember that an end behavior model lets us use a simpler function to model a more complex function as
For any polynomial function
, the function
is an end behavior model
38Slide39
Example 7: Finding End Behavior Models
Find an end behavior model for
39Slide40
Example 7: Finding End Behavior Models
Find an end behavior model for
Since
is an end behavior model for the numerator and
is an end behavior model for the denominator, then
is an end behavior model for
40Slide41
Example 7: Finding End Behavior Models
41Slide42
Example 7: Finding End Behavior Models
Find an end behavior model for
Since
is an end behavior model for the numerator and
is an end behavior model for the denominator, then
is an end behavior model for
42Slide43
Example 7: Finding End Behavior Models
43Slide44
Example 8: Finding End Behavior Models
Let
. Show that
is a right end behavior model for
while
is a left end behavior model for
.
44Slide45
Example 8: Finding End Behavior Models
Use the definition:
In the second example, evaluating
gives
. This is called an
indeterminate form
and we have not yet developed methods for finding such limits, so we will trust the graph for now.
45Slide46
Example 8: Finding End Behavior Models
46Slide47
Example 8: Finding End Behavior Models
47Slide48
Example 9: “Seeing” Limits as
Find
.
48Slide49
Example 9: “Seeing” Limits as
Find
.
We can prove that, for nearly all of the functions we will be using, if
then
What this means is that we can find the limit, then evaluate that value in
. This holds true for limits at infinity. Therefore
49Slide50
Example 9: “Seeing” Limits as
Find
.
Another way to approach this is to note that,
(which we can als0 prove). Now, if
This means that
50Slide51
Example 9: “Seeing” Limits as
51Slide52
Exercise 2.2
Online exercise 2.2
52