handa (nh5757) – Limits at infinity – bormashenko – (54880)
1
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printout
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15
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before answering.
001
10.0 points
Determine if the limit
lim
x
→ ∞
3
x
+ 2
x
2

x
+ 4
exists, and if it does, find its value.
1.
limit = 2
2.
limit = 0
correct
3.
limit =

3
4.
limit = 4
5.
limit =
2
4
6.
limit doesn’t exist
Explanation:
Dividing in the numerator and denominator
by
x
2
, the highest power, we see that
3
x
+ 2
x
2

x
+ 4
=
3
x
+
2
x
2
1

1
x
+
4
x
2
.
On the other hand,
lim
x
→ ∞
1
x
=
lim
x
→ ∞
1
x
2
= 0
.
By Properties of limits, therefore, the limit
exists and
limit = 0
.
002
10.0 points
A certain function
f
is known to have the
properties
lim
x
→ ∞
f
(
x
) = 1
,
lim
x
→ ∞
f
(
x
) = 2
.
Determine if
lim
x
→
0+
2 + 5
x
3 +
f
(
1
x
)
exists, and if it does, compute its value.
1.
limit =
7
4
2.
limit = 1
3.
limit =
1
2
4.
limit does not exist
5.
limit =
2
5
correct
Explanation:
The properties of
f
ensure that
lim
x
→
0

f
1
x
=
lim
x
→ ∞
f
(
x
) = 1
,
while
lim
x
→
0+
f
1
x
=
lim
x
→ ∞
f
(
x
) = 2
.
By properties of limits, therefore,
lim
x
→
0

2 + 5
x
3 +
f
(
1
x
)
=
1
2
,
while
lim
x
→
0+
2 + 5
x
3 +
f
(
1
x
)
=
2
5
.
Consequently,
lim
x
→
0+
2 + 5
x
3 +
f
(
1
x
)
=
2
5
.
003
10.0 points
Find all asymptotes of the graph of
y
=
3
x
2

10
x
+ 3
3
x
2

11
x
+ 6
.
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handa (nh5757) – Limits at infinity – bormashenko – (54880)
2
1.
vertical:
x
=
2
3
,
3
,
horizontal:
y
= 1
2.
vertical:
x
=
2
3
,
horizontal:
y
= 1
cor
rect
3.
vertical:
x
= 3
,
horizontal:
y
= 1
4.
vertical:
x
=
2
3
,
3
,
horizontal:
y
=

1
5.
vertical:
x
= 3
,
horizontal:
y
=

1
6.
vertical:
x
=
2
3
,
horizontal:
y
=

1
7.
vertical:
x
=

2
3
,
horizontal:
y
= 1
Explanation:
After factorization
y
=
(3
x

1)(
x

3)
(3
x

2)(
x

3)
.
Thus
y
is not defined at
x
= 3, but for
x
= 3
y
=
3
x

1
3
x

2
;
notice, however, that
lim
x
→
3
3
x

1
3
x

2
=
8
7
exists, so the graph does not have a vertical
asymptote at
x
= 3. Since
3
x

1
3
x

2
→ ∞
as
x
→
2
3
from the left and the right, the line
x
=
2
3
will, however, be a vertical asymptote.
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 Spring '10
 Gualdini
 Limits, Limit, lim, Limit of a function

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