Graphics Reference
In-Depth Information
lim
f
(
x
)
l
and lim
f
(
x
)
l
.
Give alternative definitions using sequences or neighbourhoods.
101 Use a computer or a graphics calculator to examine the graph of
1
x
f
(
x
)
4
.
Construct formal definitions for
lim
f
(
x
)
, lim
f
(
x
)
, lim
f
(
x
)
,
lim
f
(
x
)
.
Summary
-
continuity by neighbourhoods and limits
Theorem
qns 67,
68
The neighbourhood definition of continuity
A function
f
:
A
R is continuous at
a
A
if and
only if, given
0, there exists a
such that
when
x
A
and
x
a
, then
f
(
x
)
f
(
a
)
.
Theorem
qn 69
If a function
f
is continuous at a point
a
, then it
is bounded in some neighbourhood of
a
.
Theorem
qn 70
If a function is continuous at a point
a
and
f
(
a
)
0, then
f
is positivein some
neighbourhood of
a
.
Definition
qns 73, 76
One
-
sided limits
If (
f
(
a
))
l
whenever (
a
)
a
with
a
a
, then
wewritelim
f
(
x
)
l
, and say that
f
(
x
) tends
to
l
as
x
tends to
a
from below.
If (
f
(
a
))
l
whenever (
a
)
a
with
a
a
, then
wewritelim
f
(
x
)
l
, and say that
f
(
x
) tends
to
l
as
x
tends to
a
from above.
Theorem
qns 83, 84,
85
lim
f
(
x
)
l
if and only if given
0, there
exists a
such that when
x
is in thedomain of
f
and
a
x
a
, then
f
(
x
)
l
.
lim
f
(
x
)
l
if and only if, given
0, there
exists a
such that when
x
is in thedomain of
f
and
a
x
a
, then
f
(
x
)
l
.
Definition
Two
-
sided limits
lim
f
(
x
)
l
when both lim
f
(
x
)
l
and lim
f
(
x
)
l
.
