Math 234 SS 2008
CH 14.2. Limits and Continuity in Higher dimensions.
Lecture #13
Review (functions of one variable).
Limits and Continuity.
We begin with a review of the concepts of limits and continuity for
realvalued functions of one variable.
Definition 1.
Let
:
.
f
D
R
R and
let
a
R
̮�
Then
(
29
lim
x
a
f
x
L
D
=
means that for
0
ε
2200
0
δ
5
such that if
,
x
a
δ

<
then
(
29
f
x
L
ε

<
.
From the definition easily follow two fundamental results about
the limits.
1). If
c
R
D
, then
lim
x
a
c
c
D
=
2)
lim
x
a
x
a for
a
R
D
=
2200
The basic facts used to compute limits are contained in the
following theorem.
Theorem 1.
Basic Limits Theorem.
Let
,
:
.
f
g
D
R
R
̮
Suppose
(
29
(
29
lim
lim
x
a
x
a
f
x
L and
g x
M
D
=
=
. Then
(
29
(
29
(
29
(
29
(
29
(
29
(
29
(
29
0
1.lim
2. lim
3.lim
0.
x
a
x
a
x
a
f
x
g x
L
M
f
x
g x
L M
f
x
L
provided M
g x
M
D
D
D
+
=
+
=
�
�
=
1
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Math 234 SS 2008
CH 14.2. Limits and Continuity in Higher dimensions.
Lecture #13
Theorem 2.
The Squeeze (Sandwich)Theorem
.
If
(
29
(
29
lim
lim
x
a
x
a
f
x
L
g x
D
=
=
and if
(
29
(
29
(
29
f
x
h x
g x
D
,
then
(
29
lim
.
x
a
h x
L
D
=
From these assertions follows that for any polynomial or
rational function (recall that a rational function is a quotient of two
polynomials)
(
29
(
29
lim
x
a
Q x
Q a
D
=
for
Q
a
D
2200
.
Continuity
was defined taking a hint from above result.
Definition 2.
Let
:
.
f
D
R
R and
let
a
D
̮�
Then
f
is
continuous at
a
means
(
29
(
29
lim
.
x
a
f
x
f
a
D
=
By the comments preceding
Definition 2
each rational function is
continuous at each point in its domain. The same is true for all of
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 Spring '08
 Kadyrova
 Continuity, Derivative, Limits, lim, Inverse function

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