Graphics Reference
In-Depth Information
x
63
2
1
[
x
]
2
2.
There is no
such that
x
2
[
x
]
2
1or
or
.
64
(i) Any
0.08.
(ii) Any
1.
(iii) Any
0.03.
65
(i)
x
3
x
3
ยท
x
3
x
3
/
x
3
/(
2
3),
since
2
x
3.
(ii) Take
min
(
2
3), 1
.
66
(i) Given
0, there exists an
N
such that
n
N
a
3
.
(ii) Given
0, there exists a
such that
x
3
x
3
.
(iii) Given
0, there exists an
N
such that
n
N
a
3
,
which, by (ii)
a
3
.
67
(i) Given
0, there is an
N
such that
n
N
f
(
a
)
f
(
a
)
.
(ii) Given
0, there is a
such that
a
a
f
(
a
)
f
(
a
)
.
(iii) Given
0, there is an
N
such that
n
N
a
a
.
68
(i)
1/
n
does not ensure that
f
(
x
)
f
(
a
)
for all
x
satisfying
x
a
. So for at least one
x
in this
-neighbourhood of
a
f
(
x
)
f
(
a
)
.
a
) is a null sequence by qns
3.34, the squeeze rule, and 3.33, the absolute value rule.
(iii) (
a
(ii)
a
a
1/
n
ensures that (
a
)
a
. But
f
(
a
)
f
(
a
)
means (
f
(
a
)) does not tend to
f
(
a
).
69 There is a
such that
x
a
f
(
x
)
f
(
a
)
1
f
(
a
)
1
f
(
x
)
f
(
a
)
1.
70 Choose
f
(
a
).
There is a
such that
x
a
f
(
x
)
f
(
a
)
f
(
a
)
f
(
a
)
f
(
x
)
1
f
(
a
).
When
f
(
a
)
0, choose
f
(
a
).
71
x
2
, where
3/2
2
,
4/3
2
,
6/4
2
,
7/5
2
,
min(
,
,
,
).
72
(a) If
a
(0, 1] is rational and
a
a
2/
n
, then (
a
) is eventually in
thedomain of thefunction and
f
(
a
)
0. So (
a
)
a
and
f
(
a
).
(b) (i) By Archimedean order there is an integer
m
(
f
(
a
))
0
.
(ii) For each
q
, there are
q
possiblevalus for
p
and so not more
than 1
1/
2
...
m
m
(
m
1) rational numbers between
0 and 1 with denominator
m
.
