Supplemental Exercises for Section 2.4
The formula lim
x
→
0
sin
x
x
= 1 can be made more useful by observing that if
lim
x
→
a
f
(
x
) = 0 and
f
(
x
) is never 0, then lim
x
→
a
sin
f
(
x
)
f
(
x
)
= 1
.
For example
1. lim
x
→
0
sin
x
2
x
2
= 1
.
2. lim
x
→
0
sin
x
2
x
= lim
x
→
0
sin
x
2
x
2
lim
x
→
0
x
2
x
= 1
·
0 = 0
.
3. lim
x
→
1
sin(
x
2

x

2)
x
+ 1
= lim
x
→
1
sin(
x
2

x

2)
(
x
2

x

2)
lim
x
→
1
(
x
2

x

2)
x
+ 1
= 1
·
lim
x
→
1
(
x
+ 1)(
x

2)
x
+ 1
=

3
.
4. lim
x
→
1
sin(1

√
x
)
x

1
= lim
x
→
1
sin(1

√
x
)
1

√
x
1

√
x
x

1
= 1
·
lim
x
→
1
(1

√
x
)(1 +
√
x
)
(1 +
x
)(1 +
√
x
)
=
.
lim
x
→
1
1

x
(
x

1)(1 +
√
x
)
=

1
2
.
Find each of the following limits.
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This note was uploaded on 10/03/2011 for the course MTH 132 taught by Professor Kihyunhyun during the Spring '10 term at Michigan State University.
 Spring '10
 KIHYUNHYUN

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