Graphics Reference
In-Depth Information
-1
1
Figure6.11
54 If (
a
)
a
is a sequence in
A
G
then (
a
)
a
is a sequence in
A
so
(
f
(
a
))
f
(
a
) since
f
is continuous at
a
, and
(
g
(
a
))
g
(
a
) since
g
is continuous at
a
.
Also (
f
(
a
)/
g
(
a
))
f
(
a
)/
g
(
a
)
since
g
(
a
)
0 by qn 3.67, the quotient rule. So
f
/
g
is continuous at
a
.
55
(i)
f
(
x
)
f
(
x
0)
f
(
x
)
f
(0)
f
(0)
0.
(ii) 0
f
(0)
f
(
x
x
)
f
(
x
)
f
(
x
)
f
(
x
)
f
(
x
).
(iii) Let (
a
)
a
. Then (
a
a
) is a null sequence, so
(
f
(
a
a
))
f
(0), since
f
is continuous at 0.
So (
f
(
a
)
f
(
a
))
0, and so (
f
(
a
))
f
(
a
), which establishes
continuity on
R
.
(iv)
f
(1)
a
f
(1
1)
a
a
, so by induction
f
(
n
)
na
for
n
N
.
Now use(i) and (ii). 2·
f
(
x
/
n
)
f
(2
x
/
n
) etc.
(
p
/
q
)
a
.
(vi) Already proved that
f
(
x
)
ax
for rational
x
. Suppose
x
is
irrational. ([10
x
]/10
)
x
. But
f
is continuous so
(
f
([10
x
]/10
))
f
(
x
). Now
f
([10
x
]/10
)
a
[10
x
]/10
and
(
a
[10
x
]/10
)
ax
.So
f
(
x
)
ax
.
(v)
q
·
f
(
p
/
q
)
f
(
q
(
p
/
q
))
f
(
p
)
pa
f
(
p
/
q
)
57 1.8
5.
Neither of these implications may be reversed.
x
2.2
3.24
x
4.84
3
x
58 2.9
x
3.1
8.41
x
9.61
8
x
10.
59 5/3
x
7/3
3/7
1/
x
3/5
4/10
1/
x
6/10.
60 1.98
x
2.02
1.407
x
1.422
0.008
x
2
0.008.
62 Any
2 for thefirst qustion.
Any
1 for thescond, third and fourth.