steele (pss669) – HW 1 – cepparo – (55660)
1
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18
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001
10.0 points
When
f, g, F
and
G
are functions such that
lim
x
→
1
f
(
x
) = 0
,
lim
x
→
1
g
(
x
) = 0
,
lim
x
→
1
F
(
x
) = 2
,
lim
x
→
1
G
(
x
) =
∞
,
which, if any, of
A.
lim
x
→
1
g
(
x
)
G
(
x
) ;
B.
lim
x
→
1
f
(
x
)
g
(
x
)
;
C.
lim
x
→
1
f
(
x
)
g
(
x
) ;
are NOT indeterminate forms?
1.
none of them
2.
C only
correct
3.
B only
4.
A and B only
5.
all of them
6.
B and C only
7.
A and C only
8.
A only
Explanation:
A. Since
lim
x
→
1
=
∞
·
0
,
this limit is an indeterminate form.
B. Since
lim
x
→
1
f
(
x
)
g
(
x
)
= 0
0
,
this limit is an indeterminate form.
C. By properties of limits
lim
x
→
1
f
(
x
)
g
(
x
) = 0
·
0 = 0
,
so this limit is not an indeterminate form.
002
10.0 points
Use L’Hospital’s Rule to determine which of
the inequalitites
A.
100
x
< e

x
,
B.
e
2
x
< xe
x
+ 100,
C.
e
x
< x
2
+ 100,
holds for all large
x
.
1.
A and B only
2.
none of them
correct
3.
B only
4.
B and C only
5.
A and C only
6.
all of them
7.
A only
8.
C only
Explanation:
The notion of
limit at infinity
tells us that
if
lim
x
→ ∞
f
(
x
)
g
(
x
)
=
∞
,
then
f
(
x
)
g
(
x
)
>
1
for all large
x
. But then
f
(
x
)
> g
(
x
)
steele (pss669) – HW 1 – cepparo – (55660)
2
holds for all large
x
so long as
g
(
x
)
>
0 for
large
x
, allowing us to multiply through the
inequality by
g
(
x
).
Similarly, if
lim
x
→ ∞
f
(
x
)
g
(
x
)
= 0
,
then
f
(
x
)
g
(
x
)
<
1
for all large
x
, so if
g
(
x
)
>
0 for all large
x
,
then
f
(
x
)
< g
(
x
)
for all large
x
.
On the other hand, if
lim
x
→ ∞
f
(
x
) =
∞
=
lim
x
→ ∞
g
(
x
)
,
then we can use L’Hospital’s Rule to deter
mine
lim
x
→ ∞
f
(
x
)
g
(
x
)
.
Similarly, if
lim
x
→ ∞
f
(
x
) = 0 =
lim
x
→ ∞
g
(
x
)
,
then we can use L’Hospital’s Rule to deter
mine
lim
x
→ ∞
f
(
x
)
g
(
x
)
.
In this way, we can use L’Hospital’s Rule to
compare the rates of growth or decay of
f
(
x
)
and
g
(
x
) when
x
→ ∞
.
For the three given inequalities, therefore,
we have to choose appropriate
f
and
g
and
make sure that
g
(
x
)
>
0 for all large
x
.
A.
FALSE: set
f
(
x
) =
e

x
,
g
(
x
) =
100
x
.
Then
lim
x
→ ∞
f
(
x
) = 0 =
lim
x
→ ∞
g
(
x
)
,
and by applying L’Hospital’s Rule we see that
lim
x
→ ∞
f
(
x
)
g
(
x
)
= 0
.
Thus the inequality
100
x
> e

x
,
not
100
x
< e

x
,
holds for all large
x
.
B.
FALSE: set
f
(
x
) =
e
2
x
,
g
(
x
) =
xe
x
+ 100
.
Then
lim
x
→ ∞
f
(
x
) =
∞
=
lim
x
→ ∞
g
(
x
)
,
and by applying L’Hospital’s Rule twice we
see that
lim
x
→ ∞
f
(
x
)
g
(
x
)
=
∞
.
Thus the inequality
e
2
x
> xe
x
+ 100
,
not
e
2
x
< xe
x
+ 100
,
holds for all large
x
.
C.
FALSE: set
f
(
x
) =
e
x
,
g
(
x
) =
x
2
+ 100
.
Then
lim
x
→ ∞
f
(
x
) =
∞
=
lim
x
→ ∞
g
(
x
)
,
and by applying L’Hospital’s Rule twice we
see that
lim
x
→ ∞
f
(
x
)
g
(
x
)
=
∞
.
Thus the inequality
e
x
> x
2
+ 100
,
not
e
x
< x
2
+ 100
,
steele (pss669) – HW 1 – cepparo – (55660)
3
holds for all large
x
.