Graphics Reference
In-Depth Information
31 Provided that
a
,
b
0, show that
(i)
a
(e
)
e
.
(ii) log
a
x
· log
a
.
(iii) log
b
log
b
· log
a
.
Exponential and logarithmic limits
32 (
Euler
, 1748 and
Cauchy
, 1821) Usedel'Ho ˆ pital's ruleto find the
limit of
log(1
ax
)
x
as
x
0
.
Deduce that (
n
log(1
.
Usethecontinuity of
E
to show that ((1
a
/
n
)
)
e
as
n
.
a
/
n
))
a
as
n
33 Construct an argument similar to that of qn 32 to show that
a
n
e
.
lim
1
34 Thefunction
f
is defined on R
by
f
(
x
)
x
where
a
is a real
number different from 0. Prove that
f
(
x
)
a
·
x
.
35 Usetheratio tst to provethat (
n
e
)
0as
n
(qn 3.74).
Deduce that
x
as
x
and illustrate this with graph drawing facilities on a computer.
Provethat, when
a
e
0as
x
. Provethat e
/
x
0, log
x
/
x
0as
x
.
36 Investigate the function given by
f
(
x
)
, defined for positive
x
.
Show that
f
is continuous throughout its domain. Find the
minimum valueof
f
. Show that
f
is monotonic decreasing on the
domain 0
x
1/e. Show that (
f
(1/
n
))
1as
n
. Deduce that
lim
x
f
(
x
)
1.
37 What is the Maclaurin series for
E
(
x
)? Does it converge to the
valueof thefunction for all valus of
x
?
38 Usetheequation
t
)
1
(
t
)
t
)
t
)
1
(
(
...
(
t
)
,
1
(
