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Midterm Examination Math 1A Professor J. Harrison Fall 2009
Solutions
(1) (2 pts) Where is the function
f
(
x
) = ln(1 + sin(
x
)) continuous?
Answer: The function is de±ned whenever 1 + sin(
x
)
>
0. That is, we
need sin(
x
)
>

1. Since

1
≤
sin(
x
)
≤
1, we know sin(
x
)
>
1 every
where except where
sin
(
x
)=

1. This only occurs when
x
=3
kπ/
2
,
for
k
=1
,
2
,
3
, ...
. It is continuous whenever it is de±ned since sin and ln are
continuous. That is,
f
is continuous except at
x
= (3
π/
2) + (2
πk
) for
k
,
2
,
3
, ...
. (Note. You could have written much less than this, as long
as you got the ±nal answer and justi±ed it somehow.)
(2) (2 pts) Evaluate lim
x
→∞
√
3+
cos
2
x
x
2
.
Answer: lim
x
→∞
√
3=
√
3. Since 0
≤
cos
2
x
x
2
≤
1
/x
2
and lim
x
→∞
1
/x
2
=
0, we may apply the squeeze theorem to conclude
cos
2
x
x
2
= 0. By the
summation limit law, we know lim
x
→∞
√
cos
2
x
x
2
=
√
3.
(3) (2 pts) Evaluate lim
x
→∞
9
x
5

2
x
2
+1
x
4

3
x
3
+2
x
Answer: lim
x
→∞
9
x
5

2
x
2
+1
x
4

3
x
3
+2
x
= lim
x
→∞
9
x
5
x
4

2
x
2
x
4
+
1
x
4
x
4
x
4

3
x
3
x
4
+
2
x
x
4
= lim
x
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This note was uploaded on 06/26/2010 for the course MATH 1A taught by Professor Wilkening during the Fall '08 term at University of California, Berkeley.
 Fall '08
 WILKENING

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