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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.
 
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