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Why
?
Theorem
Let
fixe
and
gixs
be
two
functions
defined
in
a
neiborhood
of
x0
·
suppose
that
III
Ifix
=
A
,
(2)
9IX)=
B
,
and
BI
O
.
Then
=
Proof
Step
1
,
We
wan t
to
us
the
definition
of
limit
to
show
(3)
,
i
.
e
we
want
to
show
that
HSC0
,
=
&
(depending
on
3)
so
that
for
all
x
with
</X-Xolcd
,
we
have
(4)
I
-
<
we
want
to
find
such
&
by
using
the
definition
of
limit
for
(1)
and
(2)
.
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111
>
15)
F
9
,
30
,
&
=
d
,
(9
,
)
S
.
t
.
Ifix)
-
Al
<G
,
for
all
x
with
ocIX-Xol
&,
(2)
<)
(6)
F
&
=
>0
,
&
=
(9n)
S
-
t
.
191x
-
B1
< d2
for
all
x
with
ocIX-Xol
&
To
show
141
,
we
use
triangle
inequality
.
1
-
i
=
1
-
+
1 -
->
I
-
)+
-
=
,
/fix
-
Al
+
is
,
191x-B1
.
)
Step
2
.
Since
191x31
appears
in
the
denominator
of
formula
in
R)
,
we
wan t
to
give
a
bound
of
/gixs)
by
the
definition
(6)
as
follows
.
We
further
get
a
bound
of
ex
T
by
using
the
fact
"
to"
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(Bound
of
19151)
:
by
using
16)
We
take
92
=
I
,
then
=
&2)
=
:
Es
so
that
4
we
denote
it
by
3
18719(x)
-
B1
<
B
for
all
x
with
ocIX-XoK
&
>
Note
that
we
have
the
triangle
inequality
(19xx1
- 1B1)
=
/91x)
-
B1
.
(9)
Us ing
18)
and
(9)
together
,
for
X
with
oc1X-xold3
,
we
have
119(x1-1B1)
<
1B1
i
.
e
.
-
I
<
19xxl
-
1B1
<
i
.
2
-
B
<19(x))
<
(B)
.
(10)
<Bound
of
15xx))
:
Since
B
FO
by
the
given
condition
,
so
(10)
implies
that
for
X
with
0
<IX-Xol<
&
3
,
=>
g
By
(11)
3)
Bl
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Step
3
.
We
apply
(11)
to
(7)
and
find
that
x
Ifixs
-
Al
+
B
:
191-B1
<H
-
Al
+
19x
-
B1
-
,
Ifx
-
Al
+
2A1
+
2023
/g(x)-
B)
.
(12)
IB12
We
are
now
ready
to
show
the
definition
(4)
.
F
3 <0
,
by
using
(5)
and
taking
Ei
=
=
-1
we
find
&
,
19)
=
:
5
,
we
call
it
(i)
so
that
(13)
(f(x)
-
A)
<
9
,
= 2
for
all
x
with
<1X-xol<di
.
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By
using
(6)
again
and
taking
Ez
=
E
11312
21
Al
+
2023
we
find
diCE2)
=
:
2
(We
call
it
&2)
so
that
114)
19(x)
-
BK
<
9)
=
11312
21A1
+
2023
for
all
x
with
0
</X-xol<
&2
.
Now
we
take
8
=
mind
&
,
J
,
&33
.
Then
for
all
X
with
o
<IX-xo)
<
&
,
we
have
15
-
***
/
-
A
+
2A1
+
2023
/g(x)
-
B)
IB12
(13)
,
(14)
2
11312
< 51
.
=
+
2
A
1
+
20239
/
1B12
I'
21A)
+
2023
=
E
.
so
we
checked
(4)
the
definition
of
limit
of
quotients
.
The
proof
of
(3)
is
finished
.
17
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Remark
.
①
The
essential
part
of
the
proof
is
the
inequality
()
,
where
we
could
show
that
:
its
,
/fix
-
Al
+
it is
,
19x
-B1
bounded
Ismall
bounded
small
Once
we
have
above
idea
,
we
could
write
the
proof
.
②
The
above
is
a
regions
proof
,
where
we
use
(6)
TWICE
:
*
one
time
is
for
the
control
of
the
bound
of
Igixs)
.
*
the
other
time
is
for
control
of
the
smallness
of
191x-B1
.
③
The
TRICK
in
(12)
"21A)+2023"
is
because
it
is
NOT
ZERO
So
we
can
put
it
into
denominator
of
En
.
The
number
2025
is
just
a
positive
number
for
that
you
could
remember
the
trick
.
