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(ii) Also decide what value of
makes
f
(
x
)
1.000
correct to three places of decimals
equivalent to
f
(
x
)
1
.
(iii) So far as you can tell using your calculator, does
0
x
f
(
x
)
1
0.5?
Can you find a
such that 0
x
f
(
x
)
1
0.5?
Find a
such that 0
x
f
(
x
)
1
0.05.
Find a
x
f
(
x
)
such that 0
1
0.005.
In qn 9.27 wewill provethat lim
f
(
x
)
1
lim
f
(
x
), for the
function of qn 81 but, despite qn 79, the function
f
is not continuous at
x
0 because it is not defined there.
82
A removable discontinuity
. By choosing a valuefor
f
(0) show that it
is possible to have a function defined at
x
0 for which
lim
f
(
x
)
1
lim
f
(
x
),
but which is still not continuous at
x
0.
Definition of one-sided limits byneighbourhoods
83 A real function
f
is defined in a neighbourhood of the point
a
. If,
given any
such that, when
a
x
a
,
f
(
x
)
l
, provethat
0, there exists a
lim
f
(
x
)
l
.
Theproof may beconstructed likethat in qn 67.
84 Provethat if
lim
f
(
x
)
l
,
it follows that, given
0, there exists a
such that, when
a
.
Obtain this proof, by contradiction, by supposing that the sequence
definition of limit holds, while the neighbourhood definition fails
for some
, and finding an
a
x
a
,
f
(
x
)
l
such that
a
a
a
1/
n
and
f
(
a
)
l
, and examining the sequences (
a
) and (
f
(
a
)). The
idea of this proof is used in qn 68.