AP CALCULUS AB Chapter 2: Limits and Continuity Section 2.2: Limits Involving Infinity.
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Transcript of AP CALCULUS AB Chapter 2: Limits and Continuity Section 2.2: Limits Involving Infinity.
AP CALCULUS AB
Chapter 2:Limits and Continuity
Section 2.2:Limits Involving Infinity
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 functions for numbers large in absolute value.
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.
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
[-6,6] by [-5,5]
Example Horizontal Asymptote
Use a graph and tables to find a lim and b lim .
c Identify all horizontal asymptotes.
1
x xf x f x
xf x
x
a lim 1
b lim 1
c Identify all horizontal asymptotes. 1
x
x
f x
f x
y
Section 2.2 – Limits Involving Infinity To find Horizontal
Asymptotes:Divide numerator and denominator by the highest power of x.
Note:
2
1
002
001
112
531
lim
12
53
lim
12
53lim :Ex
32
3
333
3
33
2
3
3
3
23
xx
xx
xxx
xx
xxx
xx
xx
xx
x
x
x
01
lim xx
Example Sandwich Theorem Revisited
The sandwich theorem also holds for limits as .
cosFind lim graphically and using a table of values.
x
x
x
x
The graph and table suggest that the function oscillates about the -axis.
cosThus 0 is the horizontal asymptote and lim 0
x
x
xy
x
Properties of Limits as x→±∞
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→±∞
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 function is the constant times the limit
of the function.
5.
x
x
f x g x L M
k f x k L
Qu
: lim , 0
The limit of the quotient of two functions is the quotient
of their limits, provided the limit of the denominator is not zero.
x
f x Lotient Rule M
g x M
Product Rule:
Constant Multiple Rule:
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.
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
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
Example Vertical Asymptote
The values of the function approach to the left of 2.
The values of the function approach + to the right of 2.
The values of the function approach + to the left of 2.
The values of the funct
x
x
x
2 22 2
2 22 2
ion approach to the right of 2.
8 8lim and lim
4 48 8
lim and lim4 4
So, the vertical asymptotes are 2 and 2
x x
x x
x
x x
x xx x
2
Find the vertical asymptotes of the graph of ( ) and describe the behavior
of ( ) to the right and left of each vertical asymptote.
8
4
f x
f x
f xx
[-6,6] by [-6,6]
Section 2.2 – Limits Involving Infinity To find vertical asymptotes:1. Cancel any common factors in the numerator and
the denominator2. Set the denominator equal to 0 and solve for x.
The vertical asymptote is x=-1. (from denominator)There is a hole at x=2. (from the cancelled factor)The x-intercept is at x=-2. (from numerator)
21
22
2
4 :Ex
2
2
xx
xx
xx
x
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
Example End Behavior Models
2
2
Find an end behavior model for
3 2 5
4 7
x xf x
x
2
2
2
2
Notice that 3 is an end behavior model for the numerator of , and
4 is one for the denominator. This makes
3 3= an end behavior model for .
4 4
x f
x
xf
x
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
End Behavior Models3
In this example, the end behavior model for , is also a horizontal 4
asymptote of the graph of . We can use the end behavior model of a
rational function to identify any horizontal asymptote.
f y
f
A rational function always has a simple power function as
an end behavior model.
Example “Seeing” Limits as x→±∞
0
0
1 cosThe graph of = is shown.
1lim lim
1lim lim
x x
x x
xy f
x x
f x fx
f x fx
We can investigate the graph of as by investigating the 1
graph of as 0.
1Use the graph of to find lim and lim
1for cos .
x x
y f x x
y f xx
y f f x f xx
f x xx
Section 2.2 – Limits Involving Infinity Definition of Infinite Limits:
A limit in which f(x) increases or decreases without bound as x approaches c is called an infinite limit.
Section 2.2 – Limits Involving Infinity
c
Exit)not (Does
)(lim So
)(lim
)(lim
DNExf
xf
xf
cx
cx
cx
Section 2.2 – Limits Involving Infinity
c
)(lim So
)(lim
)(lim
xf
xf
xf
cx
cx
cx
Section 2.2 – Limits Involving Infinity Properties of Infinite LimitsIf 1. Sum or difference:
2. Product:
3. Quotient:
Lxgxfcxcx
)(lim and )(lim
Lxgxfcx
)()(lim
0 if )()(lim
0 if )()(lim
Lxgxf
Lxgxf
cx
cx
0)(
)(lim xf
xgcx