Graphics Reference
In-Depth Information
98
A squeeze rule
If
f
(
x
)
g
(
x
)
h
(
x
), for all
x
a
, provethat
g
(
x
)
l
max(
f
(
x
)
l
,
h
(
x
)
l
), when
x
a
.
Deduce that if lim
f
(
x
)
l
and lim
h
(
x
)
l
then lim
g
(
x
)
l
.
There is one important theorem about continuous functions which
does not carry over into a theorem about limits. If the function
g
is
continuous at thepoint
a
and thefunction
f
is continuous at thepoint
g
(
a
), then the function
f
g
defined by (
f
g
)(
x
)
f
(
g
(
x
)) is continuous at
thepoint
a
, thecompositerulefor continuous functions.
This is equivalent to the proposition about limits that if
lim
g
(
x
)
g
(
a
) and lim
f
(
y
)
f
(
g
(
a
))
then
lim
f
(
g
(
x
))
f
(
g
(
a
)).
However if
lim
g
(
x
)
m
and lim
f
(
y
)
l
it does not necessarily follow that
lim
(
f
g
)(
x
)
l
.
99 Consider the functions
f
and
g
defined by
g
(
x
)
0 for all
x
, and,
f
(
x
)
0.
Let
a
0, then
m
0 and
l
1 with thenotation of theprevious
paragraph.
Determine the function
f
g
and illustratethediMculty of applying
theorems about limits to composite functions.
1 when
x
0 and
f
(0)
Limits as
x
and when
f
(
x
)
100 Use a computer or a graphics calculator to examine the graph of
x
x
1
.
f
(
x
)
Construct a formal definition for
