2.2

Limits Involving Infinity

Quick ReviewIn Exercises 1 – 4, find f – 1 , and graph f, f – 1, and y = x in the same

viewing window.

32 .1 xxf xexf .2

2

31 x

xf xxf ln1

Quick ReviewIn Exercises 1 – 4, find f – 1 , and graph f, f – 1, and y = x in the same

viewing window.

xxf 1tan .3 xxf 1cot .4

xxf tan1 xxf cot1

, by ,3 3 2 2

0, by 1,

Quick ReviewIn Exercises 5 and 6, find the quotient q (x) and remainder r (x)

when f (x) is divided by g(x).

543 ,132 .5 323 xxxgxxxxf

,3

2xq

3

7

3

53 2 xxxr

Quick ReviewIn Exercises 5 and 6, find the quotient q (x) and remainder r (x)

when f (x) is divided by g(x).

1 ,12 .6 2335 xxxgxxxxf

,122 2 xxxq 22 xxxr

Quick ReviewIn Exercises 7 – 10, write a formula for (a) f (– x) and (b) f (1/x).

Simplify when possible

xxf cos .7

xxf cos

xxf

1cos

1

xexf .8

xexf

xex

f11

x

xxf

ln .9

x

xxf

ln

xxx

f ln1

xx

xxf sin1

.10

xx

xxf sin1

xxx

xf

1sin

11

What you’ll learn about Finite Limits as x→±∞ Sandwich Theorem Revisited Infinite Limits as x→a End Behavior Models Seeing Limits as x→±∞

Essential QuestionHow can limits be used to describe the behavior offunctions 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

Example Horizontal Asymptote1. Use the graph and tables to find each:

x

xxf

1

xfax lim )( 1 xfb

x lim )( 1

(c) Identify all horizontal asymptotes. 1y

Example Sandwich Theorem RevisitedThe sandwich theorem also hold for limits as x →

values.of tablea using andy graphicall cos

lim Find .2x

xx

The function oscillates about the x-axis.Therefore y = 0 is the horizontal asymptote.

0cos

lim x

xx

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

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→±∞

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 Asymptote3. Find the vertical asymptotes of the graph of f (x) and describe the

behavior of f (x) to the right and left of each vertical asymptote.

24

8

xxf

The vertical asymptotes are:

x = – 2 and x = 2

The value of the function approach –to the left of x = – 2 The value of the function approach +to the right of x = – 2 The value of the function approach +to the left of x = 2 The value of the function approach –to the right of x = 2

22 4

8lim

xx

22 4

8lim

xx

22 4

8lim

xx

22 4

8lim

xx

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

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

Example End Behavior Models4. Find the end behavior model for:

74

5232

2

x

xxxf numerator. for the modelbehavior endan is 3 2x

r.denominato for the modelbehavior endan is 4 2x

.for modelbehavior endan 4

3

4

3 makes This

2

2

xfx

x

. of asymptote horizontal thealso is 4

3 line The xfy

We can use the end behavior model of a ration function to identify any horizontal asymptote.

A rational function always has a simple power function as an end behavior model.

Example “Seeing” Limits as x→±∞ We can investigate the graph of y = f (x) as x → by

investigating the graph of y = f (1/x) as x → 0.

of lim and lim find to1

ofgraph the Use.5 xfxfx

fyxx

.1

cosx

xxf

xfy

1 ofgraph The

x

xcos

xfx lim

xf

x

1lim

0

xfx lim

xf

x

1lim

0

Pg. 66, 2.1 #1-47 odd