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Construct such a sequence in
A
and deduce that
g
(
a
)
0, so
the right-hand side of the equation is 0. Let
f
(
g
(
x
))
f
(
g
(
a
))
x
a
F
(
x
)
, for
x
a
.
Let (
a
) be any sequence tending to
a
, with
a
a
, in the
domain of
f
g
. We seek to show
(
F
(
a
))
0
f
(
g
(
a
))· 0
f
(
g
(
a
))·
g
(
a
)as
n
.
Each
a
is either in
A
or not.
If there is a subsequence of terms of (
a
)in
A
, usethe
argument for
G
, above, to show that for these
a
,(
F
(
a
))
0.
If there is a subsequence of terms of (
a
) not in
A
, usethe
algebra of part (i) and the product rule for sequences to show
that, for these
a
))
f
(
g
(
a
)) ·
g
(
a
)
0.
Deduce that, in any case, (
F
(
a
,(
F
(
a
)) is a null sequence, and that
this establishes the required limit.
Differentiability and continuity
19 Sktch thegraph of thereal function
f
given by
f
(
x
)
x
. For this
function find
f
(
h
)
f
(0)
h
f
(
h
)
f
(0)
h
lim
and lim
.
Deduce that
f
is not differentiable at
x
0.
20 (
Cauchy
, 1821) Define the function
f
:
R
R
by
f
(
x
)
sin(1/
x
) when
x
0,
f
(0)
k
.
