Infinite Limits and Limits to Infinity: Horizontal and Vertical Asymptotes.

Infinite Limits and Limits to Infinity: Horizontal

and Vertical Asymptotes

Recall…

• The notation tells us how the

limit fails to exist by denoting the unbounded behavior of f(x) as x approaches c.

• Infinity is not a number!

lim ( )x cf x

Properties of Infinite Limits

• Let c and L be real numbers and let f and g be functions such that

and

1. Sum or difference:

Consider:

lim ( )x cf x

lim ( )

x cg x L

lim[ ( ) ( )]x c

f x g x

2

1( )f x

x ( ) 2g x

0lim[ ( ) ( )]x

f x g x

0

lim ( )xf x

0lim ( )xg x

2



and

1. Product: if L > 0

if L < 0

Consider:

lim ( )x cf x

lim ( )

x cg x L

lim[ ( ) ( )]x c

f x g x

2

1( )f x

x ( ) 2g x

0lim[ ( ) ( )]x

f x g x

0

lim ( )xf x

0lim ( ) 2xg x

lim[ ( ) ( )]x c

f x g x



and

1. Quotient:

Consider:

lim ( )x cf x

lim ( )

x cg x L

( )lim

( )x c

g x

f x

2

1( )f x

x ( ) 2g x

0

( )lim

( )x

g x

f x

0lim ( )xf x

0lim ( ) 2xg x

0

0

Definition - Vertical Asymptotes

• If f(x) approaches infinity (or negative infinity) as x approaches c from the left or the right, then the line x = c is a vertical asymptote of the graph of f.

verticalasymptote

Determining Infinite Limits

3( )f x

x

2

4( )g x

x

3

2( )h x

x

0lim ( )x

f x

0lim ( )x

f x

0lim ( )x

g x

0lim ( )x

g x

0lim ( )x

h x

0lim ( )x

h x

The pattern…

Is c even or odd?

Sign of

p(x) when

x = c

odd positive

odd negative

even positive

even negative

0lim ( )x

f x 0

lim ( )x

f x

( )( ) , where ( ) is a polynomial

c

p xf x p x

x

and c is a positive integer

Using the pattern…

0

3limx

x

x

3

20

2 3limx

x x

x

50

2 1limx

x

x

0

3limx

x

x

3

20

2 3limx

x x

x

50

2 1limx

x

x

Using the pattern…

6

100

3 6limx

x

x

2

20

3limx

x x

x

6

100

3 6limx

x

x

2

20

3limx

x x

x

0

3limx

x

x

0

3limx

x

x

Limits at Infinity

• denotes that as x

increases without bound, the function value approaches L

• L can have a numerical value, or the limit can be infinite if f(x) increases (decreases) without bound as x increases without bound

lim ( )x

f x L

Horizontal Asymptotes

• The line y = L is a horizontal asymptote of f if

or

• Notice that a function can have at most two HORIZONTAL asymptotes (Why?)

lim ( )x

f x L

lim ( )x

f x L

4

2

-2

-4

-5 5

lim ( ) ______x

f x

lim ( ) ______x

f x

0 0

Horizontal Asymptote(s):__________

4

2

-2

-4

-5 5

Note: It IS possible for a graph to cross its horizontal asymptote!!!!!!

lim ( ) ______x

f x

lim ( ) ______x

f x

2 2


4

2

-2

-4

-5 5

lim ( ) ______x

f x

lim ( ) ______x

f x

0 1


lim ( ) ______x

f x

lim ( ) ______x

f x

0 0


8

6

4

2

-2

-10 -5 5 10

Theorem – Limits at Infinity

1. If r is a positive rational number and c is any real number, then

The second limit is valid only if xr is defined when x < 0

limrx

c

x lim

rx

c

x

lim x

xe

lim x

xe

0

00

0

Using the Theorem

5limx x

13

5limx x

5lim 2x x

lim x

xe

lim 2 x

xe

0 0

2

0 0

5lim lim 2x xx

lim 2 lim x

x xe

Guidelines for Finding Limits at ±∞ of Rational Functions

1. If the degree of the numerator is ___________ the degree of the denominator, then the limit of the rational function is ___.

2. If the degree of the numerator is _______ the degree of the denominator, then the limit of the rational function is the __________________ _______________________.

3. If the degree of the numerator is ___________ the degree of the denominator, then the limit of the rational function _______________.

greater than

0

less than

equal to

the ratio of the leading coefficients

is infinite

Using the Guidelines…3

4 2

2 3lim

3x

x x

x x x

3

3 2

2 9 3lim

7 1x

x x

x x

8

8 2

6 12 17lim

18 13 24x

x x

x x

5

3 2

5 6lim

2 8x

x x

x x

0

2

1

3 ∞