Tran, Tony – Homework 3 – Due: Sep 11 2007, 3:00 am – Inst: Samuels
1
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printout
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have
16
questions.
Multiplechoice questions may continue on
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before answering.
The due time is Central
time.
001
(part 1 of 3) 10 points
(i) Determine the value of
lim
x
→
5+
x

6
x

5
.
1.
limit =
∞
correct
2.
limit =
∞
3.
limit =
6
5
4.
limit =

6
5
5.
none of the other answers
Explanation:
For 5
< x <
6 we see that
x

6
x

5
<
0
.
On the other hand,
lim
x
→
5+
x

5 = 0
.
Thus, by properties of limits,
lim
x
→
5+
x

6
x

5
=
∞
.
002
(part 2 of 3) 10 points
(ii) Determine the value of
lim
x
→
5

x

6
x

5
.
1.
limit =
6
5
2.
limit =
∞
correct
3.
limit =
∞
4.
none of the other answers
5.
limit =

6
5
Explanation:
For
x <
5
<
6 we see that
x

6
x

5
>
0
.
On the other hand,
lim
x
→
5

x

5 = 0
.
Thus, by properties of limits,
lim
x
→
5

x

6
x

5
=
∞
.
003
(part 3 of 3) 10 points
(iii) Determine the value of
lim
x
→
5
x

6
x

5
.
1.
limit =
∞
2.
none of the other answers
correct
3.
limit =

6
5
4.
limit =
∞
5.
limit =
6
5
Explanation:
If
lim
x
→
5
x

6
x

5
exists, then
lim
x
→
5+
x

6
x

5
=
lim
x
→
5

x

6
x

5
.
But as parts (i) and (ii) show,
lim
x
→
5+
x

6
x

5
6
=
lim
x
→
5

x

6
x

5
.
Consequently,
lim
x
→
5
x

6
x

5
does not exist
.
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Tran, Tony – Homework 3 – Due: Sep 11 2007, 3:00 am – Inst: Samuels
2
keywords:
limit, left hand limit, right hand
limit, rational function,
004
(part 1 of 1) 10 points
Suppose that
f
(
x
) is defined for all
x
in
U
= (1
,
2)
∪
(2
,
3)
and that
lim
x
→
2
f
(
x
) =
L.
Which of the following statements is then
true?
I) If
L >
0, then
f
(
x
)
>
0 on
U
.
II) If
f
(
x
)
>
0 on
U
, then
L
≥
0.
III) If
L
= 0, then
f
(
x
) = 0 on
U
.
1.
I, II
only
2.
II only
correct
3.
II, III
only
4.
each of I, II, III
5.
None of these
6.
I, III only
Explanation:
I) False: consider the function
f
(
x
) = 1

2

x

2

.
Its graph is
2
4
6
so
lim
x
→
2
f
(
x
) = 1
.
But on (1
,
3
2
) and on (
5
2
,
3) we see that
f
(
x
)
<
0.
II) True:
if
f
(
x
)
>
0 on
U
, then on
U
the graph of
f
always lies above the
x
axis.
So as
x
approaches 2, the point (
x, f
(
x
)) on
the graph approaches the point (2
, L
). Thus
L
≥
0; notice that
L
= 0 can occur as the
graph
2
4
6
2
of
f
(
x
) =

x

2

shows.
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 Fall '08
 schultz
 Limit, lim, Mathematical analysis, Continuous function

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