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x
x
b
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7
x
t
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2
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(D)￥ ? ? ?B? ￡?á V ?￥
? Ki b h ApSE
x
xfxf
x
)()(
lim
0
→
x
fxf
x
)0()(
lim
0
=
→
x
fxf
x
+
→
)0()(
lim
0
)0(2)0()0( fff ′=′+′=
5 i b K
E5p )0(f ′
è
!
≤?
>
=
0),1(
2
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xe
x
x
x
xf
x
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2
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+
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+
x
x
x
f
x
i
2
1
lim
2
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sin
0
ππ
=
=′
→
x
e
f
x
x
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+
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yN )(xf 0=x ) V± 2
1 )(xf ),( +∞?∞
)) V± b
<
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3
xe
x
x
x
x
x
x
xf
x
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π
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fxf
x
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x
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T? ?? www.tsinghuatutor.com è 82378805
3
2008
￡
!·° Iù￡?$? bê
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0
2
lim
cos1
lim)(lim
2
000
==
=
+++
→→→
x
x
x
x
xf
xxx
y1 μ?
0)(lim)(lim
2
00
==
→→
xgxxf
xx
)(xg
yN # 0)0()(lim
0
==
→
fxf
x
)(xf 0=x ) ? ? b I ° 0=x )￥P· ?
^?i b
0)(lim
)0()(
lim
00
==
→→
xxg
x
fxf
xx
x
fxf
x
)0()(
lim
0
+
→
0
2
lim
cos1
lim
2/3
2
00
==
=
++
→→
x
x
xx
x
xx
yN D (i OM? b?
^ )0(
+
′f
)0(
_
f ′ )(xf 0=x ) V? O 0)0( =′f
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0
x )(xf
0
x
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0
x
0
x
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!f
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}￡ ~ ? 0
0
)0()(
lim)0(
0
>
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x
fxf
f
x
5 ? K ? ? Vw
i 0>δ
P ? )0,( δ?∈x ),0( δ∈x
H 0
0
)0()(
>
x
fxf
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é b
è
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x
xf
x
x
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0
)
sin
)(cos1
1(lim
5
b )0(f ′ =
"
b #
b $
0 1 2 b %
e b
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x
xf
x
x
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+
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0
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sin
)(cos1
1(lim V[? ?? 0→x
H
μ
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T? ?? www.tsinghuatutor.com è 82378805
4
2008
￡
!·° Iù￡?$? bê
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1)
sin
)(cos1
1ln(
1
lim
0
=
+?
→
x
xf
x
x
1
sin
)(cos11
lim
0
=
→
x
xf
x
x
y1 1)0( <f 5Aμ )0(0)(lim
0
fxf
x
==
→
?
^ 1
)(
lim
2
1
sin
)(cos11
lim
2
2
00
==
→→
x
xf
x
xf
x
xx
? V¨
ArE 5
?y 1 i
[ )0(f ′
2
)(
lim
)(
lim)]0([
00
2
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xf
x
xf
f
xx
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è
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H 0)( ≠xf X ? 2)0(,0)0( =′= ff
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x
x
xf
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1
0
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x
xf
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x
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n K
x
x
x
xf
x
xf
xx
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sin
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lim[2
sin
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lim
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x
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x
xf
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?
^
4
sin
1
0
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→
=? exf
x
x
b
? ? ¨?
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t K
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x
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000
xfxxfxf+=? V[ V
U1
xxxxfxfxxf+′=+ )()()()(
000
α
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6; I
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!·° Iù
T? ?? www.tsinghuatutor.com è 82378805
5
2008
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!·° Iù￡?$? bê
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? )( x?α 1
H ￥í kl
b ?:0→?x xxx=? )()( αβ 5 )( x?β
^ ￥ú¨ í k
l
b?
^? V :1
x?
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00
xxxfxfxf?+?′=? β
?l
!f
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# ×
=μ?l b ?i è
0
x A
P¤f
9
y? V
[V 1 )( xoxAy?+?=?
? AD í1
^
H ￥ú¨ í k l
5?f
x? )( xo? 0→?x )(xfy = ? ) V±
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0
x
xAdy
x
=
0
)(
0
xdf= bf
￥± sYè:1 dxxfdxydydf )(′=′==
M M
q
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)(
)()(
limlim
00
00
xf
y
x
x
xfxxf
y
x
x
x
y
y
xx
′=
+
=
→?→?
1 ￥ MM
q b y x
ü6
???l 3 pf
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q1
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Q
p
E
d
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3
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Q
p
E
d
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D±s
T
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D±s
15
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]
)(
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[
2
xg
xgxfxgxf
xg
xf ′?′
=′
E )(d)(d))()(( xgxfxgxf ±=±
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T? ?? www.tsinghuatutor.com è 82378805
6
2008
￡
!·° Iù￡?$? bê
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E E(x)d)((x) d)())()(( fxggxfxgxf += (x)d))(( fcxcf = c1è
)(
)(d)((x)d)(
)(
)(
2
xg
xgxffxg
xg
xf
d
=
])()()()[()()(])()()([ ′′+′+′=′++ xhxgxfxhxgxfxhxgxf
)()()()()()()()()( xhxgxfxhxgxfxhxgxf ′+′+′=
è
! ?i?)(xyy = ),( +∞?∞∈x
@ )(
1
2
xox
x
y
y?+?
+
=? ?
4
)1(
π
π ey = 5
=)3(y
3
π
π e b
}3 ~ ?X??
T ?i ),( +∞?∞∈x ? ?5μ )(xf ),( +∞?∞
= V? O
2
1 x
y
y
+
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2
1 x
dx
y
dy
+
= sa¤?
Cxy lnarctanln += ?
x
Cey
arctan
=
?
4
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π
π ey = ¤ π=C b?
^
x
ey
arctan
π=
3
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π
π ey = b
è
!f
()xf ( )+∞∞?,
μ ?l u W [ ]2,0
() ( )4
2
= xxxf
? ?i￥ ?
@x () ( )2+= xkfxf ? 1 è
b k
1
￥Vr
T ()xf [ )0,2?
2 ù 1′
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? '02 <≤? x 220 <+≤ x
H
() ( )2+= xkfxf ()()[ ] ( )( )42422
2
++=?++= xxkxxxk
?5
!? (),00 =f
()
() () ( )
,4
4
lim
0
0
lim0'
2
00
=
=
=
++
→→
+
x
xx
x
fxf
f
xx
()
() ()
0
0
lim0'
0
=
→
x
fxf
f
x
( )( )
k
x
xxkx
x
8
42
lim
0
=
++
=
→
7 ¤ b'() ()00
''
+?
= ff
2
1
=k
H ( )xf 0=x ) V? b
ú¨?
9
> ?
6; I
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T? ?? www.tsinghuatutor.com è 82378805
7
2008
￡
!·° Iù￡?$? bê
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?Tf
￥?f
)(xfy = )(xy′ ˉμ?
])([ ′′ xf 5 ? ])([ ′′ xf 1 ￥= ¨?
:1
)(xfy =
2
2
),(,
dx
yd
xfy ′′′′
2
2
dx
fd
b
B?ü ￥?
? 1 ￥ ¨?
:1)(
)1(
xf
n?
)(xf n
n
n
nn
dx
yd
xfy ),(,
)()(
n
n
dx
fd
b
n¨?
?l1
x
xfxxf
xf
nn
x
n
+
=
→?
)()(
lim)(
)1()1(
0
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f
￥=¨#= ¨[
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d? ú¨?
b
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b
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x
ey =
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2
L
xx
eyey
λλ
λλ =′′=′
xnn
ey
λ
λ=
)(
è p ￥ò ¨?
b xy sin=
}3 ~
+==′
2
sincos
π
xxy ?
+=
+=′′
2
2sin
2
cos
ππ
xxy
l l l l l )
2
sin(
)(
π
nxy
n
+= b
?
?f
D e é?ú ¨?
H�/
? ? L
!
(μ n¨ ?
? 1 ??
DG
T
)(xf )(xg )(xf )(xg
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0
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xgxfCxgxf
kknk
n
n
k
n?
=
∑
=
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!)!(
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k
knnn
kkn
n
C
k
n
+
=
=
L
i? b )()(
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xfxf = )()(
)0(
xgxg =
è ￥ ¨?
^ xxy sin
2
=
}3 ~ ? ??
D G
T? i? ￥ ¨?
(1
, 5μ
2
x 3≥n
)99(2)100(2)100(
)(sin)(100)(sin xxxxy ′+=
)98(2
)(sin)(
!2
99100
xx ′′
×
+
++
+=
2
99
sin200
2
100
sin
2
ππ
xxxx
+×+
2
98
sin99100
π
x
> ?
6; I
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!·° Iù
T? ?? www.tsinghuatutor.com è 82378805
8
2008
￡
!·° Iù￡?$? bê
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b xxxxx sin9900cos200sin
2
=
è p
22
1
ax
y
= ￥ ¨?
b n
}3 ~
)(
)(
22
)(
11
2
11
n
n
n
axaxaax
y
+
=
=
+
=
++ 11
)(
!)1(
)(
!)1(
2
1
n
n
n
n
ax
n
ax
n
a
+
=
++ 11
)(
1
)(
1
2
!
)1(
nn
n
axaxa
n
è ￥ ¨?
^
b )32ln()( xxf?=
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10
10
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!103
x?
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10
10
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10
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10
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a ' V b
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p?E5D±sE
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x )();( ufyx
dx
du
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?
)(uf
du
dy
′= 5ˉf
)]([ xfy?= ? )9 μ?
O x
dx
du
du
dy
dx
dy
= )()(})]([{ xufxf ′?′=′
è
x
ay
1
arctan
= D )1ln(
2
++= xxy ￥?
b
}3 ~ )(
1
1
ln)(
2
2
1
arctan
1
arctan
+
=′ x
x
aaa
xx x
a
x
a
1
arctan
2
1
ln
+
=
)2
12
1
1(
1
1
])1[ln(
22
2
x
xxx
xx
+
+
++
=′++
1
1
2
+
=
x
è pf
￥ ?
b )0(
sin
>= xxy
x
}3 ~ ZE
? ?f
?S
a·f
b
n5
H |
¤?f
b xxy lnsinln =
? ?f
p? E¤
x
x
xxy
y
sin
lncos
1
+=′ V 7 ]
sin
ln[cos
sin
x
x
xxxy
x
+=′ b
? ? 5 |
p?￥ZE ?S |
p?E b "
a¨?
a·f
?cμ
? y
TMe" {eZ a 7Z￥f
9
a¨ b
)(
)(
xv
xuy =
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9
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12
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13
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!·° Iù￡?$? bê
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14
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1
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2
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3
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x
x
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00
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0
x
0
x
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0
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lim)0(
0
>
=′
→
x
fxf
f
x
5 ? K ? ? Vw
i 0>δ
P ? )0,( δ?∈x ),0( δ∈x
H 0
0
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>
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x
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x
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sin
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5
b )0(f ′ =
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b #
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x
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x
x
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H
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1)
sin
)(cos1
1ln(
1
lim
0
=
+?
→
x
xf
x
x
1
sin
)(cos11
lim
0
=
→
x
xf
x
x
y1 1)0( <f 5Aμ )0(0)(lim
0
fxf
x
==
→
?
^ 1
)(
lim
2
1
sin
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lim
2
2
00
==
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x
xf
x
xf
x
xx
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?y 1 i
[ )0(f ′
2
)(
lim
)(
lim)]0([
00
2
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x
xf
f
xx
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x
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x
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n K
x
x
x
xf
x
xf
xx
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p¤ 'μ
]
sin
lim[]
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lim[2
sin
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x
x
x
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x
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4
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1
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x
x
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? )( x?α 1
H ￥í kl
b ?:0→?x xxx=? )()( αβ 5 )( x?β
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^
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xf
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x
x
xfxxf
y
x
x
x
y
y
xx
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=
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1 ￥ MM
q b y x
ü6
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q1
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p
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]
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6
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E E(x)d)((x) d)())()(( fxggxfxgxf += (x)d))(( fcxcf = c1è
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)(
)(
2
xg
xgxffxg
xg
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è
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xox
x
y
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4
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lim0'
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00
=
=
=
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x
xx
x
fxf
f
xx
()
() ()
0
0
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0
=
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x
fxf
f
x
( )( )
k
x
xxkx
x
8
42
lim
0
=
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=
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''
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2
1
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7
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?Tf
￥?f
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])([ ′′ xf 5 ? ])([ ′′ xf 1 ￥= ¨?
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2
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yd
xfy ′′′′
2
2
dx
fd
b
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? 1 ￥ ¨?
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n
n
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yd
xfy ),(,
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n
n
dx
fd
b
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x
xfxxf
xf
nn
x
n
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xxy ?
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2
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nxy
n
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xx ′′
×
+
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99
sin200
2
100
sin
2
ππ
xxxx
+×+
2
98
sin99100
π
x
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8
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b xxxxx sin9900cos200sin
2
=
è p
22
1
ax
y
= ￥ ¨?
b n
}3 ~
)(
)(
22
)(
11
2
11
n
n
n
axaxaax
y
+
=
=
+
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n
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n
n
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b
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2
1
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1
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x
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a
x
a
1
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2
1
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2
x
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b
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p? E¤
x
x
xxy
y
sin
lncos
1
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ln[cos
sin
x
x
xxxy
x
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p?￥ZE ?S |
p?E b "
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a·f
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9
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ZE
|
a·f
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p? E5#eE
T b
xxx
exy
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==
è
!
1
lnarctan
2
2
+
=
x
x
x
e
e
ey 5
1
1
2
1
+
=
=
e
e
dx
dy
x
b
}3 ~ |f
V r
T?1
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2
1
arctan
2
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xx
exey 5
1
1
1
2
2
2
+
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+
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x
x
x
x
e
e
e
e
y
1
1
1
2
2
2
1
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e
e
e
e
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x
1
1
2
+
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e
e
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10
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yx
yx
y
2
2
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x
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11
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1
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f
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x
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t
t
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! ptbytax sin,cos ==
x
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a
b
ta
tb
x
y
y
t
t
x
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′
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t
a
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x
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y
t
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x
ta
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ta
ta
b
32
2
sin
1
sin
sin
1
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12
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￡
!·° Iù￡?$? bê
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è
!
3
1
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+
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x
x
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f
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3
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xf
x
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x
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1
lim
000
+
+
=
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x
xf
x
xx
1
sin
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lim
1)(
lim
22
1
lim
000
x
x
x
x
xf
x
xxx
+
+
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xx
x
xf
xx
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sin
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1
00
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3
00
6
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2
1
x
x
x
xf
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lim
2
1
0
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fxf
x
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6; I
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!·° Iù
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13
2008
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lim ( )
xx
x
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fx
fx x fx
fx
x
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6; I
￡
!·° Iù
T? ?? www.tsinghuatutor.com è 82378805
14
2008
￡
!·° Iù￡?$? bê
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00
()() ( )(
()lim lim
[( ) ()] ( ) ()
lim lim ( )
xx
)f xxfx fxx fx
fx
fx x fx fx x fx
f x
→?→
→→
+ +
′?= =
′==
=
yN f
()fx￥ ?f
1
}f
b ()fx′
è ￡
ü V??
ùf
￥?f
1 μM]?
ù￥?
ùf
b
}￡ ~
!
^1 T [ 1?
ù￥ V ?f
5()fx ()(fx T fx)+ = b
00
()()()()
()lim lim ()
xx
fxT x fxT fx x fx
f xT fx
x x
→?→
++ + +
′′+= = =
()fx′' 9
^
[ T 1 ?
ù￥?
ùf
b
> ?
6; I
￡
!·° Iù
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15
