Graphics Reference
In-Depth Information
8 Show that, for thefunctions of qn 2,
f
(
x
)
f
(
x
)
1/
n
for all
x
in thedomain.
9 For thefunctions of qn 3, find valus of
x
for which
f
(
x
)
f
(
x
)
.
10 For thefunctions of qn 4, find valus of
x
for which
ยท
f
(
x
)
f
(
x
)
0.99.
11 For thefunctions of qn 5, find valus for
x
for which
f
f
(
x
)
(
x
)
.
Oneway of distinguishing between the cases in qns 1 and 2 on the
onehand and qns 3, 4 and 5 on theothr is to imaginethegraph of
thepointwiselimit functions
f
as an arm onto which sleeves of various
diameters are pulled. In qns 1 and 2, however narrow the sleeves get
the sequence of functions is eventually wholly within them. But in qns
3, 4 and 5 there are sleeves which the functions in the sequence never
get wholly inside.
12 For thefunctions in qn 3, find
sup
f
f
(
x
)
:
x
R
(
x
)
.
13 For thefunctions in qn 4, find
sup
f
(
x
)
f
(
x
)
:
x
[0, 1]
.
14 For thefunctions in qn 5, find
sup
f
(
x
)
f
(
x
)
:
x
[0,
)
.
Uniform convergence
When a sequence of functions, (
f
), with each
f
:
A
R, has a
pointwiselimit function
f
:
A
R
, and, for any
0, thegraphs of the
functions
f
are eventually inside a sleeve of radius
about thegraph of
thelimit function
f
, then, for suMciently large
n
,
f
(
x
)
f
(
x
)
f
(
x
)
for all values of
x
.
In this case we say that the sequence (
f
)
converges uniformly
to
f
.
[Think of a largefixed
n
, and check on the sleeve property by letting
x
vary right across thedomain.]
