~ ?yvD ?Dy
? 4 ?
Ki5
?×1K
~ ?yvD ?Dy
BaKi5
1.C/5
5ú ?T

nn
yx,#
n
z
@/
Hq
,lim,lim)2(
)3,2,1()1(
azay
nzxy
n
n
n
n
nnn
==
=≤≤
∞→∞→
L
*


n
x ¥Ki
 O ax
n
n
=
∞→
lim 
£,,azay
nn
→→Q
P¤,0,0,0
21
>>?>? NNε
~ ?yvD ?Dy
,
1
ε<?> ayNn
n
Hμ?
},,max{
21
NNN = |

H?,Nn>
,ε+<<ε? aya
n
'
,
2
ε<?> azNn
n
Hμ?
,ε+<<ε? aza
n


T]
H? ?,
,εε +<≤≤<? azxya
nnn
,? ?' ε<?ax
n
.lim ax
n
n
=∴
∞→

Ki¥5 V[w<?f
¥K
~ ?yvD ?Dy
5ú£ ?T?
),(
0
0
δxUx∈
Mx >
H
μ
,)(lim,)(lim)2(
),()()()1(
)()(
00
AxhAxg
xhxfxg
x
xx
x
xx
==
≤≤
∞→

∞→


*
)(lim
)(
0
xf
x
xx
∞→

i
 O?? A
?i
.
,
¥K
^ ?^ p¥Di O
Do
^/ ?¨C/5 pK1
nn
nn
zy
zy
5 ,5 ,'?1 C/5,
~ ?yvD ?Dy
è ).
1
2
1
1
1
(lim
222
nnnn
n
+
++
+
+
+
∞→
L p
3
,
1
1
1
1
2222
+
<
+
++
+
<
+ n
n
nnnnn
n
LQ
n
nn
n
nn
1
1
1
limlim
2
+
=
+
∞→∞→
?
,1=
2
2
1
1
1
lim
1
lim
n
n
n
nn
+
=
+
∞→∞→
,1=
?C/? ?¤
.1)
1
2
1
1
1
(lim
222
=
+
++
+
+
+
∞→
nnnn
n
L
~ ?yvD ?Dy
x
1
x
2
x
3
x
1+n
x
n
x
2.??μ?5
@Hq ?T

n
x
,
121
LL ≤≤≤≤
+nn
xxxx ??9F
,
121
LL ≥≥≥≥
+nn
xxxx ??h

??

5? ??μ?
AμK,
+3
d,
A M
~ ?yvD ?Dy
è
.)
(333
¥Ki
T
×?£
ü
 nx
n
+++= L
£,
1 nn
xx >
+
A ? { };
^???9¥
n
x∴
,33
1
<=xQ?
,3<
k
xL?
kk
xx +=
+
3
1
33+<,3<
{ } ;
^μ?¥
n
x∴,lim i
n
n
x
∞→

,3
1 nn
xx +=
+
Q
,3
2
1 nn
xx +=
+
),3(limlim
2
1 n
n
n
n
xx +=
∞→
+
∞→
,3
2
AA +=
2
131
,
2
131?
=
+
= AA3¤ (
 ? )
.
2
131
lim
+
=∴
∞→
n
n
x
~ ?yvD ?Dy
*3a OKi5 *
ó?¥?
i"?"¥??

P¤ε
N

l ?¥ sA1Hq
^
}{
n
x
? ?i
? 
Hüμ
Nm >
Nn>
ε<? ||
mn
xx
~ ?yvD ?Dy
A
C
=a
?×1K
(1)
1
sin
lim
0
=

x
x
x
)
2
0(,,
π
<<=∠ xxAOBO ???
!?ê?
,tan,,sin ACxABxBDx ===?
^μ
x
o
B
D
.ACO?¤T?ê?¥ ML
,xOAB¥???1
?,BDOAB¥ú1?
~ ?yvD ?Dy
,tansin xxx <<∴
,1
sin
cos <<
x
x
x'
.0
2
9? ?

T? <<
π
x,
2
0
H?
π
<< x
xx cos11cos0?=?<
2
sin2
2
x
=
2
)
2
(2
x
<
,
2
2
x
=
,0
2
lim
2
0
=

x
x
Q
,0)cos1(lim
0
=?∴

x
x
,1coslim
0
=∴

x
x
,11lim
0
=
→x
Q?
.1
sin
lim
0
=∴

x
x
x
~ ?yvD ?Dy
è,
cos1
lim
2
0
x
x
x

p
3
2
2
0
2
sin2
lim
x
x
x→
=e
T
2
2
0
)
2
(
2
sin
lim
2
1
x
x
x→
=
2
0
)
2
2
sin
(lim
2
1
x
x
x→
=
2
1
2
1
=
.
2
1
=
~ ?yvD ?Dy
(2)
e
x
x
x
=+
∞→
)
1
1(lim
?l e
n
n
n
=+
∞→
)
1
1(lim
n
n
n
x )
1
1( +=
!
L+?
+?+=
2
1
!2
)1(1
!1
1
n
nn
n
n
).
1
1()
2
1)(
1
1(
!
1
)
1
1(
!2
1
11
n
n
nnnn
++?++= LL
n
nn
nnnn 1
!
)1()1(
+
+
L
~ ?yvD ?Dy
).
1
1()
1
2
1)(
1
1
1(
)!1(
1
)
1
1
1()
1
2
1)(
1
1
1(
!
1
)
1
1
1(
!2
1
11
1
+
+
+
+
+
+
+
+
+
+
+
++=
+
n
n
nnn
n
n
nnn
n
x
n
L
L
L
,
1 nn
xx >
+
A ?
{ } ;
^???9¥
n
x∴
!
1
!2
1
11
n
x
n
++++< L
1
2
1
2
1
11
++++<
n
L
1
2
1
3
=
n
,3<
{ } ;
^μ?¥
n
x∴
.lim i
n
n
x
∞→
∴ e
n
n
n
=+
∞→
)
1
1(lim:1 )71828.2( L=e
?
1,
~ ?yvD ?Dy
,1
H? ≥x,1][][ +≤≤ xxxμ
,)
][
1
1()
1
1()
1][
1
1(
1][][ +
+≤+≤
+
+
xxx
xxx
)
][
1
1(lim)
][
1
1(lim)
][
1
1(lim
][1][
xxx
x
x
x
x
x
+?+=+
+∞→+∞→
+
+∞→
7
,e=
11][
][
)
1][
1
1(lim)
1][
1
1(lim
)
1][
1
1(lim
+∞→
+
+∞→
+∞→
+
+?
+
+=
+
+
xx
x
x
x
x
x
x
,e=
.)
1
1(lim e
x
x
x
=+∴
+∞→
~ ?yvD ?Dy
,xt?=
7
t
t
x
x
tx
+∞→?∞→
=+∴ )
1
1(lim)
1
1(lim
t
t
t
)
1
1
1(lim
+=
+∞→
)
1
1
1()
1
1
1(lim
1
+
+=
+∞→
tt
t
t
.e=
e
x
x
x
=+∴
∞→
)
1
1(lim
,
1
x
t =
7
t
t
x
x
t
x )
1
1(lim)1(lim
1
0
+=+
∞→→
.e=
ex
x
x
=+

1
0
)1(lim
~ ?yvD ?Dy
è,)
1
1(lim
x
x
x
∞→
p
3
x
x
x
∞→
+
=
)
1
1(
1
lim
1
])
1
1[(lim

∞→
+=
x
x
x
e
T
.
1
e
=
è,)
2
3
(lim
2x
x
x
x
+
+
∞→
p
3
422
)
2
1
1(])
2
1
1[(lim
+
∞→
+
+
+
+=
xx
x
x
e
T
.
2
e=
~ ?yvD ?Dy
?al2
1.
?5
2.
?×1K
C/5 ; ??μ?5,;1
sin
lim1
0
=
α
α
V?
.)1(lim2
1
0
e=α+
α
V?
,1
V??¥í kl
! α
~ ?yvD ?Dy
± I5
pK
()
x
xx
x
1
93lim +
+∞→
~ ?yvD ?Dy
± I53s
()
x
xx
x
1
93lim +
+∞→
()
x
x
x
x
x
1
1
1
3
1
9lim
+=
+∞→
x
x
x
x
x
+∞→
+?=
3
1
3
3
1
1lim9
99
0
=?= e
~ ?yvD ?Dy
.__________3cotlim4
0
=?

xx
x
a
BaA b5,
._________
sin
lim1
0
=

x
x
x
ω
a
.__________
3sin
2sin
lim2
0
=

x
x
x
a
.__________
2
sin
lim5 =
∞→
x
x
x
a
._________)1(lim6
1
0
=+

x
x
xa
5
.__________
arctan
lim3
0
=

x
x
x
a
~ ?yvD ?Dy
x
x
x
2tan
4
)(tanlim2
π

a
._________)
1
(lim7
2
=
+
∞→
x
x
x
x
a
._________)
1
1(lim8 =?
∞→
x
x
x
a
xx
x
x
sin
2cos1
lim1
0

a
x
x
ax
ax
)(lim3
+
∞→
a
=a p/
òK,
n
n
n
n
)
1
1
(lim4
2
+
+
∞→
a
~ ?yvD ?Dy
a
n
nn
n
1
)321(lim ++
∞→
?a ?¨Ki5£
ü

,......222,22,2 +++ ¥Kii p
?K
~ ?yvD ?Dy
Baa ω a
3
2
 a a
3
1

a a e a
2
e  a
e
1

=aa a
e
1
 a
a
e
2
 a
1?
e 
a
?a 2lim =
∞→
n
x
x 
5s?