3.
Trigonometric functions and in
verse trigonometric functions.
4. Exponential and logarithmic func
tions.
5. Combinations of the above.
Section 5.1: Limits of Functions
Def.
Let
f
:
D
→
R
and let
c
be an accumulation point of
D
. We
say that
lim
x
→
c
f
(
x
) =
L
if for every
>
0 there exists
δ >
0
such that

f
(
x
)

L

<
whenever
x
∈
D
and
0
<

x

c

< δ.
In other words,
lim
x
→
c
f
(
x
) =
L
means
. . .
For every
>
0,
there exists
δ >
0
such that if
x
∈
N
*
(
c, δ
)
∩
D
, then
f
(
x
)
∈
N
(
L,
).
Equivalently:
Def.
Let
f
:
D
→
R
and let
c
be an accumulation point of
D
.
A
number
L
is the
limit of
f
at
c
if for every neighborhood
V
of
L
there exists a deleted neighborhood
U
of
c
such that
f
(
U
∩
D
)
⊂
V
.
Notation
lim
x
→
c
f
(
x
) =
L
.
THEOREM 1:
Let
f
:
D
→
R
and let
c
be an accumulation point
of
D
.
If
lim
x
→
c
f
(
x
) =
L
exists, then
it is unique.
That is,
f
can have
only one limit at
c
.
Examples:
1.
lim
x
→
3
(5
x

3) = 12.
2.
lim
x
→
2
2
x
2
+ 4
x

16
x

2
= 12.
3.
lim
x
→
5
(
x
2

3
x
+ 1) = 11.
THEOREM 2:
Let
f
:
D
→
R
and let
c
be an accumulation point
of
D
. Then
lim
x
→
c
f
(
x
) =
L
if and only if
f
(
s
n
)
→
L
for every
sequence (
s
n
) in
D
such that
s
n
→
c
and
s
n
6
=
c
for all
n
∈
N
.
THEOREM 3:
Let
f
:
D
→
R
and let
c
be an accumulation point
of
D
. The following are equivalent:
1. lim
x
→
c
f
(
x
)
does not exist.
2. There exists a sequence
(
s
n
)
in
D
such that
s
n
→
c
, but (
f
(
s
n
))
does not converge.
THEOREM 4:
If
lim
x
→
c
f
(
x
) =
L,
then there exists
δ >
0
such that
f
is bounded on
N
*
(
c, δ
).
That is,
there is a number
M
such that

f
(
x
)
 ≤
M
for all
x
∈
N
*
(
c, δ
)
∩
D.
THEOREM 5:
(Arithmetic)
Let
f,
g
:
D
→
R
and let
c
be an
accumulation point of
D
. If
lim
x
→
c
f
(
x
) =
L
and
lim
x
→
c
g
(
x
) =
M,
then
1. lim
x
→
c
[
f
(
x
) +
g
(
x
)] =
L
+
M
,
2. lim
x
→
c
[
f
(
x
)

g
(
x
)] =
L

M
,
3. lim
x
→
c
[
f
(
x
)
g
(
x
)] =
LM
,
4 lim
x
→
c
[
k f
(
x
)] =
kL, k
constant,
5 lim
x
→
c
f
(
x
)
g
(
x
)
=
L
M
provided
M
6
= 0.
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 Fall '08
 Staff
 Topology, Mathematical analysis, Metric space, Limit of a sequence