Graphics Reference
In-Depth Information
value of the function at the point which seems to be in question, and
this gives a way of pinpointing discontinuity at a point. The fact that
thevalueof thefunction at thepoint in qustion dos not comeinto
the definition means that a one-sided limit may exist at a point even
when the function itself does not.
80
A jump discontinuity
. Find
x
/
x
and lim
x
/
x
.
Noticethat although thefunction
x
/
x
is not defined at
x
0 the
two limits have well-defined values.
If a function
f
: R
R is defined by
f
(
x
)
x
/
x
when
x
0, and
f
(0)
k
, useqn 79 to show that
f
is not continuous at 0 for any
choiceof
k
.
lim
There are important limits that cannot be reached which may be
interesting in their own right quite apart from questions about
continuity.
81 Let
f
bethefunction
f
: R
0
R given by
sin
x
x
.
f
(
x
)
The function is not defined at
x
0, but that does not stop there
being a limit from above, or a limit from below as
x
tends to 0.
Explore this function using a calculator. Remember to have
x
in
radians. Investigate the limit of this function as
x
tends to 0 from
above.
Usea calculator to find
f
(
x
) when
x
1,
,
,
, and continueto
find
f
(
x
) when
x
(
)
, for
n
0, 1, 2, . . . until no further change in
value takes place, because of the limits of accuracy of your
calculator.
Some calculators have a 'Fix' key, by which the number of
displayed decimal places is chosen in advance.
(i) Identify the connection between the 'Fix' key and
-neighbourhoods by deciding what value of
makes
f
(
x
)
1.00
correct to two place of decimals
equivalent to
f
(
x
)
1
.
