1 Copyright © 2014 Pearson Education, Inc. Sec. 2.3 Techniques for Computing Limits.

1Copyright © 2014 Pearson Education, Inc.

Sec. 2.3

Techniques for Computing Limits

Ex: Evaluate each limit:

2lim 5

xx

a) = 10

2 2lim lim 4

x xx

b)

= 5(2)

= 6= 2 + 4

2 2lim lim 4

x xx

c) = 8= 2 · 4

2

2

lim

lim 4x

x

x

d)4

2

1

2

3

2lim

xx

e)

3

2lim

xx

32 = 8

Sec. 2.3: Techniques for Computing Limits


2

2lim 3 7 4

xx x

a) = 30

2

3

2 4lim

3x

x x

x

b)

= 3(2)2 + 7(2) + 4

23 32 4

33

7

6



3lim 8x

x a) = 2

2

0lim 7 9

xx

b) 27 90 9

3 8

3



2lim cos

xx

a) = 0

2

3 2lim sin

xx

b) 2 3

2sin

23

sin2

2cos

1

21



2

4

12lim

4x

x x

xa)

4

4 3lim

4x

x x

x

4

lim 3x

x

4 3

7



3

1

1lim

1x

x

xb)

2

1

1 1lim

1x

x x x

x

2

1lim 1

xx x

21 1 1

3



0

4 2lim

x

x

xc)

0

4 2 4 2lim

4 2x

x x

x x

0

4 4lim

4 2x

x

x x

0lim

4 2x

x

x x

0

1lim

4 2x x

1

0 4 2

1

4 2

1

2 2

1

4


Memorize these!

Memorize these!

Memorize these!



0

tanlim

x

x

xa)

1

0

sin 1lim

cosx

x

x x

0 0

sin 1lim lim

cosx x

x

x x

1 1

"The limit of a product …"

"… equals the product of the limits."



0

sin5lim

x

x

xb)

5

0

5 sin5lim

5x

x

x

0

sin55 lim

5x

x

x

0

sin5 lim

y

y

y

5 1

Let y = 5x.

x 0 if and only if y 0

In order to use , the argument and the denominator must be the same.


1 Copyright © 2014 Pearson Education, Inc. Sec. 2.3 Techniques for Computing Limits.

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