Graphics Reference
In-Depth Information
negative number is a lower bound. 0 is both.
26
(i) satisfies (a) and (c) but tends to
.
(ii) satisfies (a) and (b) but tends to 1.
(iii) satisfies (c) and perhaps (d) but tends to
0.1 and so keeps its
distancefrom 0.
(iv) possibly satisfies (c) and (d) but keeps at least 0.001 away from
0.
(v) satisfies (e) but oscillates between 0 and 1.
27 (i)
n
10; (ii)
n
100; (iii)
n
1000; (iv)
n
1/
.
28 For qn 26 (i), any
will do. For qn 26 (ii) take
1 or less. For qn
26 (iii) take
0.1 or less. For qn 26 (iv) take
0.01 or less. For
qn 26 (v) take
1 or less.
For the constant sequence 1, 1, 1, take
. For no term in this
a
sequence is
.
29 Given
0, let
N
be an integer greater than 1/
(which exists by
the property of Archimedean order). Then
n
N
n
1/
1/
n
.
30 Given
0, let
N
be an integer greater than 1/
(which exists by
the property of Archimedean order). Then
n
N
n
1/
1/
n
.
31 Given
0, there exists an
N
such that
n
N
a
.
For given
C
0, there exists
N
such that
n
N
a
1/
0
1/
a
1/
C
0
a
1/
C
C
1/
a
, so (1/
a
)
.In
general (
a
)
0 does not imply (1/
a
)
. Put
a
(
1)
/
n
, for
example.
32 Case(i),
c
0. Given
0, choose
N
so that
n
N
a
/
c
.
Then
c
·
a
.
Case(ii),
c
0. Given
0,
c
·
a
0
, for all
n
.
Useqn 30 and put
c
10.
33 Since
a
a
, thesame
N
is suMcient for either sequence.
34 If
a
and 0
b
a
, then 0
b
, and so
b
.So
n
N
a
b
.
35
(i) Use qn 29 and the squeeze rule (qn 34).
(ii) Usethescalar rule(qn 32) and part (i).
(iii) Show that 1/(7
n
3)
1/7
n
. Then use qn 29, the scalar rule
(qn 32) and the squeeze rule (qn 34).
(iv) Show that 1/(
n
1)
1/
n
. Then use qn 29 and the squeeze
rule(qn 34).
