1.5 infinite limits. copyright © houghton mifflin company. all rights reserved. 21-2 figure 1.25

1.5 Infinite Limits

Copyright © Houghton Mifflin Company. All rights reserved. 21-2

Figure 1.25


Definition of Continuity


Figure 1.26


Figure 1.28

Determining Infinite Limits from a Graph


Theorem 1.10 The Existence of a Limit


Definition of Continuity on a Closed Interval and Figure 1.31


Theorem 1.11 Properties of Continuity


Theorem 1.12 Continuity of a Composite Function


Theorem 1.13 Intermediate Value Theorem


Figure 1.35 and Figure 1.36


Definition of Infinite Limits and Figure 1.40


Definition of Vertical Asymptote


Theorem 1.14 Vertical Asymptotes

Finding Vertical Asymptotes

• Determine all vertical asymptotes of the graph.

A) B) C)1( )2( 1)

f xx

2 1( )2 1xf xx

( ) cotf x x

Rational Function with Common Factors

2

2

2 8( )

4

x xf x

x

Determining Infinite Limits

• Find each limit.

2 2

1 1

3 3lim and lim

1 1x x

x x x xx x


Theorem 1.15 Properties of Infinite Limits

Determining Limits

• Because

20 0

20

1lim1 1 and lim , you can write

1lim 1 Property 1, Theorem 1.15

x x

x

x

x

Determining Limits

• Because

0 0

0

lim 3 3 and lim cot , you can write

lim 3cot Property 2, Theorem 1.15

x x

x

x

x

Determining Limits

• Because

2

1 1

2

1

lim 1 2 and lim cot , you can write

1lim 0 Property 3, Theorem 1.15

cot

x x

x

x x

xx

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