2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
$?s
?B [±s 
? cf
￥KD ??f

f
￥Kà
Q
f
í kù)￥K
g?z
K￥à
QDZE￥ -4/ V[
¨ ?1Dzf
￥K o31?i
f
Kù5 ú1M
￥ t_? ?[/ ? f ?
→
0
xx  
+
→
0
xx xx →  +∞→x ?∞→x  ∞→x b
g?zf
K￥à
QDZE
^éB?1Df
???a ?
?a?à
Q￥×1
$b
?l 
!f
 uW)(xfy = ),( ∞+a
=μ?l 0>?ε  ?i
?è
AD

P?
Hμ0>X Xx > ε<? Axf )( 5? )(xfy = ? t??í k v
H￥K 1x
A
l ?? Ab:1 Axf
x
=
+∞→
)(lim b
?

￥è
5 ?
^? t??í kv
H￥í kl
b ?

?l?
￥
0=A )(xf x
A?i5? ? t??í kv
H￥K?i? ?b)(xf x
?

?l￥+ilD
KP ?
b
? l 
!f
)(xfy =  u W ),( ∞+a
=μ ?l  0>?G  ?i
 ?è
AD

P?
Hμ0>X Xx > Gxf >)( 5?
^? t??í kv
H￥í kv
b
:1
)(xf x
∞=
+∞→
)(lim xf
x
b
? ?μμ ?/￥
? f ? 
 +∞=
+∞→
)(lim xf
x

DGxf >)(?∞=
+∞→
)(lim xf
x

 Gxf?<)( 
 ?

? ?l  Vó ?∞→x
H ￥K D 1 í kv
￥ ?lb
? V
ó/

)(xf )(xf
∞→x
H￥KDí kv
￥?l
í

 Axf
x
=
∞→
)(lim ∞=
∞→
)(lim xf
x
+∞=
∞→
)(lim xf
x
∞=
∞→
)(lim xf
x
b
?l 
!f
)(xfy =  uW ),( ∞+?∞
=μ?l ?i
?è
AD 
P
?
0>X
Xx >
Hμ ε<? Axf )( 5 ? )(xfy = ? t?í kv
H￥K1x A
l ?
? Ab:1 bAxf
x
=
∞→
)(lim
 ? 

￥è
5 ?
^? t ?í kv
H￥í kl
 b ?

? l ?
￥
0=A )(xf x
A?i 5? ? t?í kv
H￥K?i ? ?b ?+Y?i ? ú [)(xf x x
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L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
_Z
T t?í kvb
?
-
￥?l V[ó? 
_ t?í kv
H 1í kv
￥ ??
í
'?í kv
μí kv

_í kv
b h????t
b
x )(xf
f
B?)￥K
?l 
!f
)(xfy =  ￥ ??
#×
0
x },{),(
*
00
00
><?<= δδδ xxxxN 
=μ?l ? 0>?ε ? i
?è
AD 0
0
>δ 
δδ <
0

P ?
00
0 δ<?< xx
H
μ ε<? Axf )( 5? ? t?
H￥K1)(xfy = x
0
x A
l ?? Ab:1
 Axf
xx
=
→
)(lim
0
b
?

￥è
5 ?
^? t?
H￥í kl
b  ?

?l?￥0=A )(xf x
0
x A
?i5? ? t?
H￥K?i? ?b)(xf x
0
x
?l 
!f
 uW)(xfy = ),(
00
δ+xx 
0>δ 
=μ?l ? 0>?ε ? i
?è
AD 0
0
>δ 
δδ <
0

P?
00
0 δ<?< xx
Hμ ε<? Axf )( 5?
? t?
H￥·K1
)(xf
x
0
x A:1
 Axf
xx
=
+
→
)(lim
0
b
7?f
 uW)(xfy = ),(
00
xx δ? 
0>δ 
=μ?l ? 0>?ε ? i
 ?
è
AD 0
0
>δ 
δδ <
0

P ? 0
00
<?<? xxδ
Hμ ε<? Axf )( 5 ?
? t?
H￥PK1
)(xf
x
0
x A:1
 Axf
xx
=
→
)(lim
0
b
+Y

?? K
T ? ￥ A^1 ∞∞± 
' )(xf |′í KMv 
HsY ?
M? t_/￥í kv
?aμí kv
b:1)(xf
∞=
→
)(lim
0
xf
xx
 ±∞=
+
→
)(lim
0
xf
xx
 ±∞=
→
)(lim
0
xf
xx
b

f
Ki￥Hq
? ? K 
Axf
x
=
∞→
)(lim ∞±∞,i￥ 1Hq
^ 

D 
?i O
Axf
x
=
+∞→
)(lim ∞±∞,
Bxf
x
=
∞→
)(lim ∞±∞,BA= 
∞±∞,b
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L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
? ?  K 
Axf
xx
=
→
)(lim
0
∞±∞,i￥ 1 Hq
^ D
?i O
Axf
xx
=
+
→
)(lim
0
Bxf
xx
=
→
)(lim
0
BA= b
Ki￥5
??μ?5
? ? b
!f
 uW)(xfy = ),( δ+aa 
0>δ 
=μ?l O?? hμ/? 
5·K Axf
ax
=
+
→
)(lim ib 7?f
)(xfy =  uW ),( aa δ? 
0>δ 
=μ?l O
??9μ
?
HPK Axf
ax
=
→
)(lim ib
bC/5
? ? b
!f
D)(xfy = )(),( xxg φ  uW ),( δδ +? aa 
0>δ 
=μ?l O
@ )()()( xgxfx <<φ  ? Axxg
axax
==
→→
)(lim)(lim φ i5 ibAxf
ax
=
→
)(lim
í kl
Dμ¨f
￥e￥Ki O ˉ1í kl
b'
! 
? t_/μ? è ?  ?i
?è
)(xf 0>M
),0( ∞+∈?x ?μ Mxf ≤)(  0)(lim =
+∞→
xg
x
5 0)()(lim
0
=
→
xgxf
xx
b
? V[T1Ki￥5 ??¨b

?SK
?¨

?5 V[¤?/
?SK
×1K
SK  1
0
=
→
x
x
x
sin
lim 

SK  ex
x
x
=+
→
1
0
1)(lim  

f
K￥?é

?é
[/òH (
a¨? ∞±∞→,x ￥ f?


! C 1
Lè
5Axf
xx
=
→
)(lim
0
( ) CAxfC
xx
=?
→
)(lim
0
b


! Axf
xx
=
→
)(lim
0
Bxg
xx
=
→
)(lim
0
5 ( ) BAxgxf
xx
±=±
→
)()(lim
0



! Axf
xx
=
→
)(lim
0
Bxg
xx
=
→
)(lim
0
5 ( ) ABxgxf
xx
=?
→
)()(lim
0
b


! Axf
xx
=
→
)(lim
0
0≠)(xg  0
0
≠=
→
Bxg
xx
)(lim 5
B
A
xg
xf
xx
=
→
)(
)(
lim
0
b


! 0
0
≠∞=
→
)(,)(lim xfxf
xx
5 0
1
0
=
→
)(
lim
xf
xx
b
?¨


?é V[9


tKb 1ZL9
 ??í kv

H ?
!E|
í kv
1í kl
b


?é￥
5?
T (1 sHq ?
@ -
Hq
H 2
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L 1101 è 62781785
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￡
!·°
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L 1101 è 62781785
?B??? ?b  I
k? K￥
E5￥¨p
^ènp ?+Y?iE
? ?
pb
3?é#ˉK? ? 
f
K μBt×1￥3?é ?1 ?K￥ ?? 
 |? DˉK? ?
g??t?é) ?K[#a?￥±sDs
= ?μ?vúùb /
ó?t?é ([
￥ f?1 èb
0
xx →
? ? K￥ ??
 |? 
? 5  ￥?í
" ? 
 u W
= A ?μ bD y
- ? 5i  ￥ ??
#×
0
0
>=
→
Axf
xx
)(lim
0
x
0
x 0>)(xf
0
0
>=
→
Axf
xx
)(lim
0
x },{),(00
00
><?<= δδδ xxxxN 
P? ),( δ
0
xNx∈
HA ?μ b? ?0>)(xf 0
0
<=
→
Axf
xx
)(lim 5 ￥? í
"
? 
 uW
=A ?μ b
0
x
0
x 0<)(xf
￡? 0
0
>=
→
Axf
xx
)(lim 5 0>?ε ?i
?è
DA 0>δ 
P? δ<?<
0
0 xx
Hμ ε<? Axf )(  εε +<<? AxfA )( b
+Y | 0
2
>=
A
ε 5
2
3
)(
2
A
xf
A
<< ?
^μ 0
2
)( >>
A
xf b
?N? é V[w
 ? Axf
xx
=
→
)(lim
0
 Bxg
xx
=
→
)(lim
0
 O BA> 5i ￥
 ??
#×
0
x
},{),(00
00
><?<= δδδ xxxxN 
P? ),( δ
0
xNx∈
H
μ bi OéB?μ ?/w
)()( xgxf >
K ??￥
I
h?1?
￡
ü
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" ? 
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=  O K i5
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" ? 
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=
H5
b
0
x
0
x 0>)(xf )(lim xf
xx
0
→
0
0
≥=
→
Axf
xx
)(lim
0
x
0
x 0<)(xf
0
0
≤=
→
Axf
xx
)(lim
? ? μ?? 
 ?K i5  ￥?í
" ? 
 uW
=μ?b)(lim xf
xx
0
→
)(xf
0
x
0
x
? ? ˉK? ?
? )(,)(lim xuuAuf
uu
==
→
0

0
0
uxu
xx
=
→
)(lim 
0
xx ≠
H 5
0
uu ≠
Axuf
xx
=
→
))((lim
0


ˉ K? ? 9
a¨? ?
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b?B ? ? V[
P ¤ K9
M￥ ÷F y?
ZLb ? ¨ K
? é#ˉ  K? ? V [ ¤?K ￥ ?N
í
[#
? S K￥
M?Vr
T ?/
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L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
)()()(lim
)(
xAxfAxf
n
α+=?=
→


1=
→
)(
)(sin
lim
)(
x
x
x
α
α


()ex x
x
=+
→
)(
)(
)(lim αα
1
1  

? )(xα 1
? t_
H￥í kl
 O
D
?￥)(?→x 0≠)(xα b
è pK )(sinlim nn
n
+
∞→
22
π b
3
±
^
^μ ?b
n5? ??f
o?
T
)(sinlim nn
n
+
∞→
22
π ()
2
2
1 ππ nnn
n
n
+?=
∞→
)sin()(lim 
()
2
2
ππ nnn
n
+=
∞→
)sin(lim  )(sinlim ππ nnn
n
+=
∞→
22

?ˉK? ?o?9

)(lim ππ nnn
n
+
∞→
2
2
2
π
π =
++
=
∞→
nnn
n
n
lim 
yN 1
2
222
==+
∞→
π
π sin)(sinlim nn
n
b
í kl
1¨
?l 
! )(xα D )(xβ 1
? t_ )(?→x
H￥í kl
 ?
@
μ
β
α
=
→
)(
)(
lim
)(
x
x
x
 

5
? 0≠μ
H? )(xα D )(xβ 1]¨í kl
 
)(?→x +Y 1=μ
H? )(xα
D )(xβ 1?Ní kl

)(?→x  V:1 )(xα _ )(xβ b

? 0=μ
H? )(xα
^1 )(xβ ú¨￥í kl

)(?→x b

? ∞=μ
H? )(xα
^1 )(xβ ?¨￥í kl

)(?→x b
?¨K?é#
 V[¤?/
+Fè¨?Ní kl

0→x
 )ln(~tanx~sinx~ xx +1  

2
2
1
1 xx ~cos?  

axa
x
ln~1? 
 
0>a
xe
x
~1?  

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L 1101 è 62781785
2005
￡
!·°
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L 1101 è 62781785
xx λ
λ
~)(11?+ 
R∈λ  

3
6
1
xxx ~sin 

?
[
?N1" V<l/ ?¨'? N1"?￥ x ?¨?èD1
@
0)(lim
)(
=
→
x
x
α ￥
? )(xα b

K 
? V[¨?NNí kl
é?9D ?A??i 9Do
y0ê
?
é?y?Ní kl

^¨y0e
)(
)(
x
x
β
α
1
?l￥bdE9D
^ènpb
 ??f
à
Q
??f
￥ à
Q ?
 ?Z

n 5
^f
 B?) ?? ￥à
Q? 1
^¨ ? Yf
B? # ?í￥  ? f ?±4?
  Q
^f
 uW
 ? ? ￥à
Q ? 1
^¨ ?
Yf
 vS?
= ￥ ?  f ? 4?
b 7
μ? t à
Q?|
^ éB?ù ?f
? é
￥A!$b

f
B? ??￥à
Q
?l 
!f
)(xfy = 

 ￥

#×
0
x },{),(0
00
><?= δδδ xxxxN 
=μ?l 

K i Axf
xx
=
→
)(lim
0

 )(
0
xfA=
5?f
 ) ?? b[
 ?H n V ?1f
 B? ? ?￥ ?1
í  ? B
? Vb? ? 

=￥ ?iB?) ? ??5 ?f


= ??b
)(xfy =
0
x
)(xfy = ],[ ba ),( ba )(xfy =
],[ ba ),( ba
)(xfy =  ) ??￥°4il
^?
0
x
0
xxx?=? ?il
H
)()()(
00
xfxfxf?=? 9 V[ ?ilb
?f
7
 ?? ?￥×12
^Bí ?f
 ?l×
= ?￥ ?i u W
=
?
^ ??￥b
f
B? ??￥?l Vμ[/
??N?
í
 
?N?
í

!f
 ￥

#×)(xfy =
0
x },{),(00
00
><?<= δδδ xxxxN
=
μ?l i O
@ )()()( xxfxf α+=
0
  ? )(xα 1í kl

b5?f
 ) ??b
0
xx →
)(xfy =
0
x
?N?
í

!f
 ￥

#×)(xfy =
0
x },{),(0
00
><?= δδδ xxxxN
=μ?
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L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
l ? ?:|
0
xxx?=?  )()()(
00
xfxfxfy?=?=?  ? μ 0
0
=?
→?
y
x
lim 5?f
 ) ??b)(xfy =
0
x
[

??N
í
èè?1
f
B?) ??￥
m
b b
? l  ?
@ 5?f
 ) 1· ?? 7
@
H5?f
 )1P ??b
)()(lim
0
0
xfxf
xx
=
+
→
)(xf
0
x
)()(lim
0
0
xfxf
xx
=
→
)(xf
0
x
?
^
f
B?) ??￥ZEμ[/? ?
? ? f
 ? ) ??￥ 1 Hq
^  O
b
)(xfy =
0
x )()(lim
0
0
xfxf
xx
=
+
→
)()(lim
0
0
xfxf
xx
=
→

í
1 B?) ??￥ 1Hq
^??)P ?? O· ??b
?l f
)(xfy = ? ??￥??-1 ￥W?bW?S ?/s ? )(xf
??HK D ?i ? ? ??
H ? 1?B ? W?b  ?
@
￥W??-1 V ??W?b
)(lim xf
xx
+
→
0
)(lim xf
xx
→
0
0
x
)(lim xf
xx
+
→
0
)(lim xf
xx
→
=
0
V ??W?￥ V
f ?
^ =
→
)(lim
0
xf
xx
)()(lim
0
0
xfxf
xx
≠
+
→
í?l)(
0
xf
7
P¤ ￥?B ?W??è?1K?W?b)(lim xf
xx
+
→
0
)(lim xf
xx
→
≠
0
" ??B ?W?[?￥
μW?d?1?= ?W?b ?
P¤ ￥
??1í kW? ?
H ?μ?9 |′vl?9M |′￥??1? ?
W?b
∞=
→
)(lim xf
xx
0
0
xx → )(xf

f
B?) ??￥?é
?é  ?f
 ) ??5)(xfy =
0
x )()()(
0
xxfxf α+=  ? 0)(lim
0
=
→
x
xx
α b
?é 
 |?  ?f
 ) ??i O 5i ￥

# × )(xfy =
0
x 0
0
>)(xf
0
x
},{),(
00
><?= δδδ xxxxN ),(
P¤? δ
0
xNx∈
Hμ b0>)(xf
?é 
μ ?? ?f
 ) ??5iè
D ￥

# × )(xfy =
0
x 0>M
0
x
},{),(
00
><?= δδδ xxxxN ),(
P¤? δ
0
xNx∈
Hμ Mxf ≤)( b
' 
)(xf ),( δ
0
xN
=μ?b
??é  V[
P¤  ??f
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v 1e
M 1e?￥f
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  7 ? é 
?1 ??f
￥ ? ?  |? a?
= ? D? ? f
￥ ?
ù?[ # s￥ 
￡
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L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
??μ"×1T¨b?é 9è¨?f
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ù?￥? b

 ??f
￥
?éˉf
DQf
￥ ???
? ? b ?f
D  ) ?? 5f
)(xf )(xg
0
x )()( xbgxaf ±  
) ?? O?
Hf
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0
x
0
0
≠)(xg
)(
)(
xg
xf
 )g ??b
0
x
? ? bˉ ??? ?
?  ) ??)(ufy =
0
uu = )(xuu = 
0
xx = ) ?? O 
00
uxu =)(
5  ) ??b))(( xuf
0
xx =
? ? 
!f
> uW
?ì ?? O ?? 5)(xfy = ],[ ba )(xfy = 

￥Qf

],[ ba
)(yf
1?
],[ βα
?ì?? ??b
> uW
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￥?é
f
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Q#B "
?é
^ù?f
vS?
= ￥ ? ?
4?
￥×1
m
9
^a?D
= ?￥×1$b

? ? μ??? ?
?f
μ ?> u W
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μ? ' i

P μ
)(xfy = ],[ ba )(xf ],[ ba
0>M ][ bax∈? Mxf ≤)( b
￡Q￡bL
!?"￥ ?i5 |0>M ],[ bax
n
∈ 
P¤ ),2,1()(L=> nnxf
n
A ? μ?Aμ
l ?-0
  O 
n
x
0
x
n
x
k
→
k
k
n
n
xf >)(L,2,1=k 
?y1 ? ????b
n
xa
k
≤≤ ],[
0
bax ∈ μˉK? ?D ￥ ???
Aμ ?
^ μ?b ?D  M
±
[ 
μ?b 
)(xf
)()(lim
0
xf
n
xf
k
k
=
∞→
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k
n
xf
k
k
n
n
xf >)( )(xf
],[ ba
? ? KvKl′? ?
!f
> uW
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μKvKl′ b' i 
D
)(xfy = ],[ ba )(xf ],[ ba
],[ bax
M
∈ ],[ bax
m
∈
P¤ )(max)(
],[
xfMxf
bax
M
∈
== O )(max)(
],[
xfmxf
bax
m
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T? www.tsinghuatutor.com - 8 - bvD ? S
L 1101 è 62781785
2005
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L 1101 è 62781785
P¤ b0
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T? www.tsinghuatutor.com - 9 - bvD ? S
L 1101 è 62781785
2005
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!·°
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W￥ ?i
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e
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x
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x
4
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0
1
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1
2
4
1
0
=
+
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x
x
e
e
lim  0
1
2
1
2
4
34
0
4
1
0
=
+
+
=
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→→
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x
xx
x
x
x
x
e
ee
e
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x
x
x
e
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4
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2
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lim ?ib
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T? www.tsinghuatutor.com - 10 - b vD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
=
→
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sin
lim
x
x
x 0
1
0
=
→
x
x
x
sin
lim  1
0
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lim
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x
x

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sin
lim
x
x
x 0→
9?ib
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5 (1 sHq
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 ￥K?A?ib? ?SE ?/
 =
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+
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+
→
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sin
lim
x
x
e
e
x
x
x
4
1
0
1
2
+
+
+
+
→
x
x
x
e
e
4
1
0
1
2
lim 110
0
=+=
+
→
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lim
x
x
x

=
+
+
+
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lim
x
x
e
e
x
x
x
4
1
0
1
2
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x
x
e
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0
1
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lim 112
0
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x
x

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1
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b
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n
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o?
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n
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a
x
x
x
a

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a
x
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a
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x
a
a
x
x
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1
1
1
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￡
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T? www.tsinghuatutor.com - 11 - b vD ? S
L 1101 è 62781785
2005
￡
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ê?7<v0 1006 bvD ? S
L 1101 è 62781785
1
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ax
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e
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x
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1
2
eex
x
x
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=
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1
1
21
x
x
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lim 
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xe
x
1
1
21
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2
21
2
1
2
1

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=?= e
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xe
x
lim b
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11
2
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xx
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11
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xx
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xxx
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x
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x
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lim
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lim1lim1lim
2
22
3ln
1
2
3ln
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1
2
=
=
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+∞→
xx
x
x
x
x
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x
xn
x
x
x
x
3 ?K1o p?? I
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x
x
x
x
cos
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lim
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1
0
x
xx
xxx
x
x
x
xx
→

+=
sin
cossin
sin
lim
1
1
0
1 ?i?
e
x
xx
xx
x
x
=?

+
→
sin
sin
lim 1
0
?ˉK? ?o3 p ?/K
3
1
6
112
1
1
3
2
00
==
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sin
cos
lim x
xxx
xx
x
xx

￡
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T? www.tsinghuatutor.com - 12 - b vD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
?
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1
1
1
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x
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lim b
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2
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x
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23
1
2
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x
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2
1
12
1
11
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1
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xx
x
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2
0
2
1
x
x
x
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lim
2
1
2
2
2
0
=
+
→
x
x
x
sin
lim 
2
1
2
2
2
00
==

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x
x
xf
xx
lim)(lim
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2
1
0 =)(f 5 
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x
x
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x
x
x
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T? www.tsinghuatutor.com - 13 - b vD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
51)(coslim)(cos
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→→
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x
bx
e
x
x
x
x
¤ 4?=b b
yN 4,1?== ba
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X ? ∞≠=
→→
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xf
x
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→H Y
D G Y
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b

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k
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n
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k
n
n
n
ee
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=
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1
1
1
0
2
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2
1
2
1
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n
e
n
n
e
k
n
k
n
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x
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x)(φ
￡
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T? www.tsinghuatutor.com - 14 - b vD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
| xxf sin)( =
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xf
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xf
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23
lim
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12
0
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x
xx
x
e
ee

2
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0
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x
yNs?1 #
b 
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x
bxax
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→0
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x
x
bxax
cot
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→0
x
x
x
ax
bx
ax
cot
cot
cos
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)(coslim?
+=
→
1
0

→
→
++=
x
x
ax
bx
bx
ax
x
xax
ax
x
ax
bx
ax
sin
cos
cos
sin
sin
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lim)cos(lim 111
0
1
1
1
0

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b
è 2.15
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11
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xxx
xf
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 )1,
2
1
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k? ?l
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2
1
[
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yy
yy
xx
xx
xf
yxx
ππ
ππ
πππ
ππ
π sin
sin
lim
1
sin)1(
sin)1(
lim
1
)(lim
011
+=

+=
+
→→→
y
y
y
yy
yy
2
0
22
0
2
cos
lim
1sin
lim
1
π
πππ
ππ
ππ
π
+=
+=
++
→→
ππ
ππ
π
1
2
2
1
lim
1
2
22
0
=
+=
+
→
y
y
y
o3?l )1(f
π
1
= b
è ￡
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￡
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^7/?¨
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￡
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T? www.tsinghuatutor.com - 15 - b vD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
),( bax ∈
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0
00
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0
b
è ￡
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 uW)(xfy = ),[ ∞+a
 ? ? O i 5
 uW
μ?b
Axf
x
=
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x
=
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Hμ ?'i #
 ?

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1
>M
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Mxf ≤)( A ?f
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 ??yN
f
 uW
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2
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2
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),[ ∞+a
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 uW
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?
5?1 ??f
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)()( xfxxF?= ),( ba 0<?= )()( afaaF #
 uW
?¨
,?? ? V?f
0>?= )()( bfbbF ],[ ba )()( xfxxF?=  uW
=μ
,?b'i
P¤),( ba ),( bax ∈
0 000
0xxfxF =?= )()( b
è 
! ￡
ücba <<
cxbxax
xf
+
+
=
111
)( ?  uW
=  μ
?
L?b
),( ca
￡
n5 1 ??í kW? ?l×1cbax,,= )(xf
),(),(),(),( +∞∪∪∪?∞ ccbbaa ò0 uW
= (1 ??f
b

=),( a?∞ 0
111
<
+
+
=
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xf )( y Ní?b 7
=),( +∞c
￡
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2005
￡
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ê?7<v0 1006 bvD ? S
L 1101 è 62781785
0
111
>
+
+
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cxbxax
xf
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=à
μ B ?
L? ?
^ 
= μ B?
L
?b
)(xf ),( ba
)(xf ),( ba )(xf ),( ba

=),( cb
cxbxax
xf
+
+
=
111
)( g1 1?? hf

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Kμ B?
L? ?y1 +∞=
+
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→
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=A ?M| Q?
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L? ?
^ 
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μB?
L?b
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)(xf ),( cb )(xf ),( cb
8

s
cxbxax
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+
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111
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 ??  O 0=
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Aμ?￥Kv′b
￡ ?? 
= V | ?? ′5 à
μB?
P
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xfxf < ],[ XXx?∈ )(xf ],[ XXx
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P
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[ b
)(
m
xf
],[ XX? ],[
1
XXx?∈ 0)()(
1
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è 
! 
 ?? O
@)(xf ],[ a20 )()()( afaff ≠= 20 
k￡
üi
),( ax 0
0
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00
b
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 )()()( axfxfxF +?= 
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0200 =?=+ )()()()( affaFF yNAμ )()(FaF 0?= ? ??f
￥
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i
P¤),( ax 0
0
∈ 0
0
=)(xF ' )()( axfxf +=
00
b
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^ ￡
ü?
T
D??
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è Z? 
=
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 bxe
x
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￡
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L 1101 è 62781785
"
 #
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3
7 xexf
x
=
2
1)( 0)0( =f  0
2
1
1)
2
1
(
1
>=
ef b
?y1 ?K￥ ?? i?∞=
+∞→
)(lim xf
x
),
2
1
(
1
∞+∈x
P¤ y
Ni
0)(
1
<xf
),
2
1
(
12
xx ∈
P¤ 0)(
2
=xf b
6? 12)(
2
=′
x
exf 01
2
)
2
1
( <?=′
e
f  
[ ?04)(
2
<?=′′
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exf
),
2
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( ∞+∈x
H 01
2
)
2
1
()( <?=′<′
e
fxf  ??h
 ?
^
^ )(xf
2
x )(xf ),
2
1
( ∞+
=￥·B
L?b
? )
2
1
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H  ??9F01)( >>′ xf )(xf 0)0()( => fxf  í
L?b )(xf
8

s?
^
^ 
2
x )(xf ),0( ∞+
=￥·B
L?b
è 
! 
μ ? l()xf ∞+0( 1=x ) ??i O
@
() ( )xfxf = 
k￡ ()xf ∞+0(
1è
b
}3~ ?B
,E Vμ () ( )xfxf = )
2
1
(
n
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=L
 () )(
2
1
n
xfxf = 
7 ∞→n ∞+∈? 0(x ?ˉ K? ? ¤?
() Cfxfxfxf
n
nn
====
∞→∞→
)1()(lim)(lim
2
1
b

￡
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T? www.tsinghuatutor.com - 18 - b vD ? S
L 1101 è 62781785