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12 Limits and an Introduction to Calculus Copyright © Cengage Learning. All rights reserved.
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Find the equation for each horizontal asymptote.
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3 2
9 7 184 2 5 1
x xf xx x x
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3 2
5 7 238 3 4x xg xx x
1.
2.
3. 2
35h xx
Copyright © Cengage Learning. All rights reserved.
12Limits and an Introduction to
Calculus
12.4
Copyright © Cengage Learning. All rights reserved.
LIMITS AT INFINITY AND LIMITS OF SEQUENCES
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• Evaluate limits of functions at infinity.
• Find limits of sequences. We will do this later.
What You Should Learn 12-4
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Limits at Infinity and Horizontal Asymptotes
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Limits at Infinity and Horizontal Asymptotes
There are two basic problems in calculus: finding tangent lines and finding the area of a region.
We have seen how limits can be used to solve the tangent line problem. In this section and the next, you will see how a different type of limit, a limit at infinity, can be used to solve the area problem.
To get an idea of what is meant by a limit at infinity, consider the function given by
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Limits at Infinity and Horizontal Asymptotes
The graph of f is shown in Figure 12.30.
Figure 12.30
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Limits at Infinity and Horizontal Asymptotes
From earlier work, you know that is a horizontal asymptote of the graph of this function.
Using limit notation, this can be written as follows.
These limits mean that the value of f (x) gets arbitrarily close to as x decreases or increases without bound.
Horizontal asymptote to the left
Horizontal asymptote to the right
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Limits at Infinity and Horizontal Asymptotes
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Limits at Infinity and Horizontal Asymptotes
To help evaluate limits at infinity, you can use the following definition.
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Example 1 – Evaluating a Limit at Infinity
Find the limit.
Solution:Use the properties of limit
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Example 1 – Solution= 4 – 3(0)
= 4
So, the limit of f (x) = 4 – as x approaches is 4.
cont’d
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You Do Exercise 9, page 887
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3lim lim 1
0 1 1
x xx
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A Mr. Green time-saver:
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1 1 01
1 1 0
n nn n
m mm m
a x a x a x aR xb x b x b x b
lim limn
nx x m
m
a xR xb x
does not exist if
lim if
0 if
nn n
x mm m
n ma x a n mb x b
m n
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You Do Exercise 19, page 887
2
2 2 2
2
2 2 2
2
2
4 2 1
lim3 2 2
2 14 4 0 0lim 2 2 3 0 03
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t
t
tttt ttt
tt t
tt
tt
2
2
Mr. Green's method:
4 4lim3 3ttt
Work through how to do the graph by hand.
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-5 5
-5
5
x
yGraph of
2
_ 2
4 2 13 2 2x xyx x
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Limits at Infinity and Horizontal Asymptotes
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Anything with a bar over it means the average.
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You Do Exercise 55, page 888
This is so old! Let’s say it’s a smart phone!
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You Do Exercise 55, page 888
a)
13.50 45,750
45,75013.5
CCx
xx
x
b)
c) 45,750lim 13.5 13.5x x
As more PDAs are produced, the average cost per unit will approach $13.50.
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Exit
Why is it that when we evaluate limits toward infinity of rational functions, that we can find the limit simply by using the terms of highest degree in the numerator and the denominator?
