Graphics Reference
In-Depth Information
1
1
n
x
3
If
f
(
x
)
,
find the limit function for the sequence (
f
) on thedomain
R
.
4 If
f
(
x
)
x
, find the limit function for the sequence (
f
) on the
domain [0, 1].
x
1
5
If
f
(
x
)
1
,
x
find the limit function for the sequence (
f
) on thedomain [0,
).
Because these limit functions are determined point by point for
each
x
of thedomain of thefunctions, they arecalled
pointwise limit
functions
.
A formal definition runs like this: if a sequence of real functions (
f
),
f
:
A
R
, havethesamedomain
A
and, for each
x
A
, thelimit
lim
f
(
x
) exists, then a function
f
:
A
R may be defined by
f
(
x
)
(
x
).
This
f
is the
pointwise limit function
for the sequence (
f
lim
f
). It is important
to stress that pointwise limit functions are found by keeping
x
constant
and letting
n
.
It is somewhat disconcerting to find that even though the functions
in the sequence are continuous it is possible for the pointwise limit
function to bediscontinuous.
This chapter investigates the context in which a property which is
common to all the functions of a sequence is retained by the limit
function. Thepossibility that continuity may belost in thelimit shows
how significant this question can be. We will distinguish carefully
between the cases where the pointwise limit function is continuous, and
the cases where it is not.
6 In each of qns 1, 2, 3, 4 and 5, attempt to find a value of
x
such
that
(i)
f
(
x
)
f
(
x
)
,
(ii)
f
(
x
)
f
(
x
)
.
7 Continuing the investigation started in qn 6, show that, for the
functions of qn 1,
f
(
x
)
f
(
x
)
1/
n
for all
x
in thedomain.
