Graphics Reference
In-Depth Information
Thefunction
g
is continuous at 0 by qn 6.35. But
g
is not differentiable at
x
0 because
g
(
x
)
g
(0)
x
0
f
(
x
) when
x
0,
and
f
has no limit as
x
0.
Since0
h
(
x
)
x
for all
x
,
h
is continuous at 0 by qn 6.36.
h
(
x
)
h
(0)
x
0
g
(
x
) when
x
0, and lim
g
(
x
)
g
(0)
so
h
is differentiable at 0.
25
(i) (
f
(
x
)
f
(0))/(
x
0)
x
sin(1/
x
) which tends to 0 as
x
tends to 0, as
in qn 21. When
x
0,
f
(
x
)
2
x
sin(1/
x
)
2/
x
cos(1/
x
).
f
(1/
(2
n
1)
)
2
((2
n
1)
).
(ii)
f
(
x
)
f
(0)
x
0
1
2
x
cos
1
x
1as
x
0, as in qn 21.
1
2
n
1
(2
n
1)
1
2
n
2
4
n
1
2
n
1
2
(2
n
1)
f
f
,
2
1
4
n
2
n
2
1.
When
x
0,
f
(
x
)
1
4
x
cos(1/
x
)
2 sin(1/
x
).
When
f
is positive throughout an interval, this counter-intuitive possibility
cannot occur, as we shall see in the next chapter.
26
R
0
.
27 When
x
0,
f
(
x
)
1. When
x
0,
f
(
x
)
1.
28
f
(1)
0.
f
(
x
)
f
(1) for all
x
.So
f
(1) is theminimum valueof thefunction.
29 If for all
x
in somenighbourhood of
a
,
f
(
x
)
f
(
a
) then
f
has a
local
maximum
at
a
. If for all
x
in somenighbourhood of
a
,
f
(
x
)
f
(
a
) then
f
has a
local minimum
at
a
.
f
(
1)
f
(1)
0.
30 If
f
has a local maximum at
a
then in some neighbourhood of
a
f
(
x
)
f
(
a
)
x
a
0 when
x
a
, and
f
(
x
)
f
(
a
)
x
a
0 when
x
a
.
f
(
x
)
f
(
a
)
x
a
0 and lim
f
(
x
)
f
(
a
)
x
a
0.
Therefore lim
Since the two-sided limit exists it equals 0.
