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