1 Copyright © 2014 Pearson Education, Inc. Sec. 2.3 Techniques for Computing Limits
1 Copyright © 2014 Pearson Education, Inc. Sec. 2.3 Techniques for Computing Limits.
Jan 01, 2016
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1Copyright © 2014 Pearson Education, Inc.
Sec. 2.3
Techniques for Computing Limits
Copyright © 2014 Pearson Education, Inc. 2
Copyright © 2014 Pearson Education, Inc. 3
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
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
Sec. 2.3: Techniques for Computing Limits
Ex: Evaluate each limit:
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
Sec. 2.3: Techniques for Computing Limits
Sec. 2.3: Techniques for Computing Limits
Ex: Evaluate each limit:
3lim 8x
x a) = 2
2
0lim 7 9
xx
b) 27 90 9
3 8
3
Sec. 2.3: Techniques for Computing Limits
Sec. 2.3: Techniques for Computing Limits
Sec. 2.3: Techniques for Computing Limits
Ex: Evaluate each limit:
2lim cos
xx
a) = 0
2
3 2lim sin
xx
b) 2 3
2sin
23
sin2
2cos
1
21
Sec. 2.3: Techniques for Computing Limits
Sec. 2.3: Techniques for Computing Limits
Ex: Evaluate each limit:
2
4
12lim
4x
x x
xa)
4
4 3lim
4x
x x
x
4
lim 3x
x
4 3
7
Sec. 2.3: Techniques for Computing Limits
Ex: Evaluate each limit:
3
1
1lim
1x
x
xb)
2
1
1 1lim
1x
x x x
x
2
1lim 1
xx x
21 1 1
3
Sec. 2.3: Techniques for Computing Limits
Ex: Evaluate each limit:
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
Sec. 2.3: Techniques for Computing Limits
Copyright © 2014 Pearson Education, Inc. 17
Copyright © 2014 Pearson Education, Inc. 18
Copyright © 2014 Pearson Education, Inc. 19
Memorize these!
Memorize these!
Memorize these!
Sec. 2.3: Techniques for Computing Limits
Ex: Evaluate each limit:
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."
Sec. 2.3: Techniques for Computing Limits
Ex: Evaluate each limit:
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.
Sec. 2.3: Techniques for Computing Limits
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