Graphics Reference
In-Depth Information
Provethat, given
h
0,
f
(
a
h
)
g
(
a
h
)
l
f
(
a
h
)
(
a
h
)
l
g
for some
1.
Deduce that if 0
k
implies
, with 0
f
(
a
k
)
g
(
a
k
)
l
,
h
then 0
implies
f
(
a
h
)
g
(
a
h
)
l
,
so that
f
(
x
)
g
(
x
)
l
.
lim
It is easy to see that a result analogous to that of qn 26 can be
obtained using limits from the left if appropriate conditions apply.
When the functions
f
and
g
are differentiable at
a
, either result may
beused.
27 Usedel'Ho ˆ pital's rule, qn 26, to find
(
x
2)
2
(
x
1)
1
,
(i) lim
e
1
x
(ii) lim
sin
x
x
, hence lim
1
cos
x
x
(iii)
lim
,
x
sin
x
x
and hence lim
.
28 If, in qn 26, wehad had
f
(
x
)
g
(
x
)
, instead of lim
f
(
x
)
g
(
x
)
l
.
lim
would we have been able to deduce that
f
(
x
)
g
(
x
)
lim
?
