Horstman (mdh995) – HW10 – Radin – (58505)
1
This
printout
should
have
16
questions.
Multiplechoice questions may continue on
the next column or page – find all choices
before answering.
001
10.0 points
Compute the value of
lim
n
→∞
2
a
n
b
n
6
a
n
−
b
n
when
lim
n
→∞
a
n
= 6
,
lim
n
→∞
b
n
=
−
2
.
1.
limit doesn’t exist
2.
limit =
−
13
19
3.
limit =
−
12
19
correct
4.
limit =
12
19
5.
limit =
13
19
Explanation:
By properties of limits
lim
n
→
2
2
a
n
b
n
= 2 lim
n
→∞
a
n
lim
n
→∞
b
n
=
−
24
while
lim
n
→∞
(6
a
n
−
b
n
)
= 6 lim
n
→∞
a
n
−
lim
n
→∞
b
n
= 38
negationslash
= 0
.
Thus, by properties of limits again,
lim
n
→∞
2
a
n
b
n
6
a
n
−
b
n
=
−
12
19
.
002
10.0 points
Determine if the sequence
{
a
n
}
converges
when
a
n
=
1
n
ln
parenleftbigg
4
6
n
+ 4
parenrightbigg
,
and if it does, find its limit.
1.
the sequence diverges
2.
limit = ln
2
5
3.
limit = ln
2
3
4.
limit =
−
ln 6
5.
limit = 0
correct
Explanation:
After division by
n
we see that
4
6
n
+ 4
=
4
n
6 +
4
n
,
so by properties of logs,
a
n
=
1
n
ln
4
n
−
1
n
ln
parenleftbigg
6 +
4
n
parenrightbigg
.
But by known limits (or use L’Hospital),
1
n
ln
4
n
,
1
n
ln
parenleftbigg
6 +
4
n
parenrightbigg
−→
0
as
n
→ ∞
. Consequently, the sequence
{
a
n
}
converges and has
limit = 0
.
003
10.0 points
Determine if the sequence
{
a
n
}
converges,
and if it does, find its limit when
a
n
=
3
n
5
−
n
3
+ 4
7
n
4
+ 4
n
2
+ 2
.
1.
the sequence diverges
correct
2.
limit =
−
1
4
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Horstman (mdh995) – HW10 – Radin – (58505)
2
3.
limit = 0
4.
limit = 2
5.
limit =
3
7
Explanation:
After division by
n
4
we see that
a
n
=
3
n
−
1
n
+
4
n
4
7 +
4
n
2
+
2
n
4
.
Now
1
n
,
4
n
4
,
4
n
2
,
2
n
4
−→
0
as
n
→ ∞
; in particular, the denominator
converges and has limit 7
negationslash
=
0.
Thus by
properties of limits
{
a
n
}
diverges
since the sequence
{
3
n
}
diverges.
004
10.0 points
Determine whether the sequence
{
a
n
}
con
verges or diverges when
a
n
=
8
n
2
8
n
+ 4
−
n
2
+ 2
n
+ 1
,
and if it does, find its limit
1.
the sequence diverges
2.
limit =
1
2
correct
3.
limit =
1
4
4.
limit =
1
6
5.
limit = 0
Explanation:
After bringing the two terms to a common
denominator we see that
a
n
=
8
n
3
+ 8
n
2
−
(8
n
+ 4)
(
n
2
+ 2
)
(8
n
+ 4) (
n
+ 1)
=
4
n
2
−
16
n
−
8
8
n
2
+ 12
n
+ 4
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 RAdin
 Calculus, Squeeze Theorem, Limits, Limit, lim, Limit of a function

Click to edit the document details