Graphics Reference
In-Depth Information
1
(1
x
)
(viii)
f
(
x
)
,
(ix)
f
(
x
)
x
x
1
x
(x)
f
(
x
)
.
The first thing to look for in these graphs, are the values of
x
for which
the sequence of real numbers (
f
(
x
)) is convergent. One does this by
thinking of
x
as constant and letting
n
. Having obtained a limit
function
f
in this way, one then compares the graphs of
f
and
f
as a
whole, looking for their greatest distance apart and thinking of
n
as
constant.
Pointwise limit functions
1 Sktch thegraphs of thefunctions given by
1
1
x
1
4
x
1
9
x
f
(
x
)
,
f
(
x
)
, and
f
(
x
)
,
superimposing the diagrams on the same axes, using appropriate
computer software.
1
n
x
For a constant real number
x
, find lim
.
Because this limit is well defined for each real number
x
, wehavea
limit function f
: R
R for the sequence of functions (
f
) defined by
1
n
x
f
(
x
)
.
Thelimit function in this caseis given by
f
(
x
)
0, for all
x
.
2 Sktch thegraphs of thefunctions given by
f
(
x
)
(sin
x
)/
n
, for
n
1, 2 and 3,
using computer software.
Is thelimit lim
sin
x
/
n
well defined, for each real number
x
?
Sincelim
f
(
x
)
0 for all
x
in this case, the function
f
defined by
f
(
x
)
0 is thelimit function for thesequence(
f
).
In qns 1 and 2, thelimit functions wreconstant giving thesame
valuefor each
x
. In qns 3, 4 and 5, thevalus of thelimit arenot the
samefor all valus of
x
.
