chapter 2 2012 pearson education, inc. 2.2 limits involving infinity section 2.2 limits and...
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2012 Pearson Education, Inc.
Chapter 2
2.2Limits Involving Infinity
Section 2.2
Limits and Continuity
Slide 2.2- 2 2012 Pearson Education, Inc.
Quick Review
1 1In Exercises 1– 4, find and graph , and in the
same viewing window.
1. 2 3 2. x
f f f y x
f x x f x e
Slide 2.2- 3 2012 Pearson Education, Inc.
Quick Review
1 13. tan 4. cotf x x f x x
Slide 2.2- 4 2012 Pearson Education, Inc.
Quick Review
3 2 3
5 3 3 2
In Exercises 5 and 6, find the quotient and remainder
when is divided by .
5. 2 3 1, 3 4 5
6. 2 1, 1
q x r x
f x g x
f x x x x g x x x
f x x x x g x x x
Slide 2.2- 5 2012 Pearson Education, Inc.
Quick Review
1In Exercises 7 –10, write a formula for a and b .
Simplify where possible.
7. cos 8.
ln 19. 10. sin
x
f x fx
f x x f x e
xf x f x x x
x x
Slide 2.2- 6 2012 Pearson Education, Inc.
Quick Review Solutions
1
1
1
1In Exercises 1– 4, find and graph , and in the
same viewing window.
1.
3ln
2
2 3 2. x
f f f y x
f
xf
x e
f
f
x
x
x x
x
[12,12] by [8,8] [6,6] by [4,4]
Slide 2.2- 7 2012 Pearson Education, Inc.
Quick Review Solutions
, by ,3 3 2 2
0, by 1,
1 1
1 13. tan 4. cot
tan cot
02 2
f x x f x
f x x f
x
x
x
x
x
Slide 2.2- 8 2012 Pearson Education, Inc.
Quick Review Solutions
3 2 3
5 3 3 2
2
2 2
In Exercises 5 and 6, find the quotient and remainder
when is divided by .
5. 2 3 1, 3 4
2 5 7, 3
3 3 3
2 2 1,
5
6. 2
2
1, 1
q x r x
f x g x
f x x x x
q
g x x x
f x x x x g x
x r x
x
x x
q x x x r x x x
x
Slide 2.2- 9 2012 Pearson Education, Inc.
Quick Review Solutions
11 1 1cos , cos ,
ln 1 1,
1In Exercises 7 –10, write a formula for a and b .
Simplify where possible.
7. cos 8.
lln
9.
110. s
n
x x
x
f x x f f x e f ex x x
xf x f x
x x x
f x fx
f x x f x e
xf x
x
f x xx
1 1 1 1sin , sin ni f x x x f x
x x xx
x
Slide 2.2- 10 2012 Pearson Education, Inc.
What you’ll learn about
Finite Limits as x→±∞ Sandwich Theorem Revisited Infinite Limits as x→a End Behavior Models Seeing Limits as x→±∞
…and whyLimits can be used to describe the behavior of functionsfor numbers large in absolute value.
Slide 2.2- 11 2012 Pearson Education, Inc.
Finite limits as x→±∞
The symbol for infinity (∞) does not represent a real number. We use ∞ to describe the behavior of a function when the values in its domain or range outgrow all finite bounds.
For example, when we say “the limit of f as x approaches infinity” we mean the limit of f as x moves increasingly far to the right on the number line.
When we say “the limit of f as x approaches negative infinity (∞)” we mean the limit of f as x moves increasingly far to the left on the number line.
Slide 2.2- 12 2012 Pearson Education, Inc.
Horizontal Asymptote
The line is a of the graph of a function
if either
lim or limx x
y b
y f x
f x b f x b
horizontal asymptote
Slide 2.2- 13 2012 Pearson Education, Inc.
[-6,6] by [-5,5]
Example Horizontal Asymptote
1xf x
x
a lim 1x
f x
b lim 1x
f x
c Identify all horizontal asymptotes. 1y
c Identify all horizontal asymptotes.
Use a graph and tables to find a lim and b lim .x x
f x f x
Slide 2.2- 14 2012 Pearson Education, Inc.
Example Sandwich Theorem Revisited
The sandwich theorem also holds for limits as . x
cosFind lim graphically and using a table of values.
x
x
x
The graph and table suggest that the function oscillates about the -axis.x
cosThus 0 is the horizontal asymptote and lim 0
x
xy
x
Slide 2.2- 15 2012 Pearson Education, Inc.
If , and are real numbers and
lim and lim , then
1. : lim
The limit of the sum of two functions is the sum of their limits.
2. : lim
The limi
x x
x
x
L M k
f x L g x M
Sum Rule f x g x L M
Difference Rule f x g x L M
t of the difference of two functions is the difference
of their limits
Properties of Limits as x→±∞
Slide 2.2- 16 2012 Pearson Education, Inc.
Properties of Limits as x→±∞
3. : lim
The limit of the product of two functions is the product of their limits.
4. : lim
The limit of a constant times a
x
x
Product Rule f x g x L M
Constant Multiple Rule k f x k L
function is the constant times the limit
of the function.
5. : lim , 0
The limit of the quotient of two functions is the quotient
of their limits, provi
x
f x LQuotient Rule M
g x M
ded the limit of the denominator is not zero.
Slide 2.2- 17 2012 Pearson Education, Inc.
Properties of Limits as x→±∞
6. : If and are integers, 0, then
lim
provided that is a real number.
The limit of a rational power of a function is that power of the
limit of the function, provided the latt
rrss
x
r
s
Power Rule r s s
f x L
L
er is a real number.
Slide 2.2- 18 2012 Pearson Education, Inc.
Infinite Limits as x→a
If the values of a function ( ) outgrow all positive bounds as approaches
a finite number , we say that lim . If the values of become large
and negative, exceeding all negative bounds as x a
f x x
a f x f
approaches a finite number ,
we say that lim . x a
x a
f x
Slide 2.2- 19 2012 Pearson Education, Inc.
Vertical Asymptote
The line is a of the graph of a function
if either
lim or lim x a x a
x a
y f x
f x f x
vertical asymptote
Slide 2.2- 20 2012 Pearson Education, Inc.
Example Vertical Asymptote
[6,6] by [6,6]
Find the vertical asymptotes of the graph of ( ) and describe the behavior
of ( ) to the right and left of each vertical asymptote.
f x
f x
2
8
4f x
x
The values of the function approach to the left of 2.x The values of the function approach + to the right of 2.x The values of the function approach + to the left of 2.x The values of the function approach to the right of 2.x
2 22 2
8 8lim and lim
4 4x xx x
2 22 2
8 8lim and lim
4 4x xx x
So, the vertical asymptotes are 2 and 2x x
Slide 2.2- 21 2012 Pearson Education, Inc.
End Behavior Models
The function is
a a for if and only if lim 1.
b a for if and only if lim 1.
x
x
g
f xf
g x
f xf
g x
right end behavior model
left end behavior model
Slide 2.2- 22 2012 Pearson Education, Inc.
Example End Behavior Models
Find an end behavior model for
2
2
3 2 5
4 7
x xf x
x
2
2
Notice that 3 is an end behavior model for the numerator of , and
4 is one for the denominator. This makes
x f
x2
2
3 3= an end behavior model for .
44
xf
x
Slide 2.2- 23 2012 Pearson Education, Inc.
End Behavior Models
1
If one function provides both a left and right end behavior model, it is simply
called an .
In general, is an end behavior model for the polynomial function nn
n nn n
g x a x
f x a x a x
end behavior model
10... , 0
Overall, all polynomials behave like monomials.na a
Slide 2.2- 24 2012 Pearson Education, Inc.
End Behavior Models
3In this example, the end behavior model for , is also a horizontal
4asymptote of the graph of . We can use the end behavior model of a
rational function to identify any horizontal asymptote.
A
f y
f
rational function always has a simple power function as
an end behavior model.
Slide 2.2- 25 2012 Pearson Education, Inc.
Example “Seeing” Limits as x→±∞
We can investigate the graph of as by investigating the y f x x
1graph of as 0.
y f xx
1Use the graph of to find lim and lim
x x
y f f x f xx
1for cos .f x x
x
1 cosThe graph of = is shown.
xy f
x x
0
1lim lim
x x
f x fx
0
1lim lim
x x
f x fx
Slide 2.2- 26 2012 Pearson Education, Inc.
Quick Quiz Sections 2.1 and 2.2
2
3
You may use a graphing calculator to solve the following problems.
61. Find lim if it exists
3
A 1
B 1
C 2
D 5
E does not exist
x
x x
x
Slide 2.2- 27 2012 Pearson Education, Inc.
Quick Quiz Sections 2.1 and 2.2
2
3
You may use a graphing calculator to solve the following problems.
61. Find lim if it exists
3
A 1
B 1
C 2
E does not exist
D 5
x
x x
x
Slide 2.2- 28 2012 Pearson Education, Inc.
Quick Quiz Sections 2.1 and 2.2
2
3 1, 22. Find lim = if it exists5
, 21
5A
313
B 3
C 7
D
E does not exist
x
x xf x
xx
Slide 2.2- 29 2012 Pearson Education, Inc.
Quick Quiz Sections 2.1 and 2.2
2
3 1, 22. Find lim = if it exists5
, 21
13B
3C
5A
3
7
D
E does not exist
x
x xf x
xx
Slide 2.2- 30 2012 Pearson Education, Inc.
Quick Quiz Sections 2.1 and 2.2
3 2
3
3. Which of the following lines is a horizontal asymptote for
3 7
2 4 53
A2
B 0
2C
37
D53
E2
x x xf x
x x
y x
y
y
y
y
Slide 2.2- 31 2012 Pearson Education, Inc.
Quick Quiz Sections 2.1 and 2.2
3 2
3
3. Which of the following lines is a horizontal asymptote for
3 7
2 4 53
A2
B 0
2C
37
D
3E
5
2
x x xf x
x x
y x
y
y
y
y