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2,9 3,1f x2,9 1 3 2 3,1 1x x x1 0x
2,9 1 2 3 3,1 1 2x x x2,9 4,9 3 3,1 5,1x x x
1
3 3,1 5,1 0x x2,9 4,9 3 0x x
0,1 5,1x0,1 4,9x5,10,1
x4,90,1
xA
lim 3x
f x
3 2 53 3
1 1x
f xx x
3 0f xfC
limx
f x ylimx
f x yf x y
4lim 3x
f x
2,95 3,05f x2
2,95 3,052
xx
2,95 2 2 3,05 2 2x x x
8,12,05
x7,91,95
x3,951219512... 4,051282051...x
3,95;4,05I
2limx
f x310f x233 4 10 2x x21000 4003 3996 0x x
4003 40009 4003 400092000 2000
x1,901488751 2,101511249x
0,1a
lim 1x
f xlim 1x
f x1
limx
f x1
limx
f x
limx
f xlimx
f x2
limx
f x2
limx
f x
lim 4x
f xlim 4x
f x3
limx
f x3
limx
f x
lim 0x
f xlim 0x
f x
lim 0x
f xlim 0x
f x2
limx
f x
limx
f xlimx
f x0
limx
f x
limx
f xlimx
f x0
limx
f x0
limx
f x
lim 3x
f xlim 3x
f x1
limx
f x
limx
f xlimx
f x3
limx
f x3
limx
f x
lim 0x
f xlim 0x
f x1
limx
f x
1limx
f x4
limx
f x4
limx
f x
limx
f xlimx
f x1
limx
f x
1lim
xf x
3limx
f x3
limx
f x
limx
f xlimx
f x2
limx
f x
limx
f x
0x
21
2lim lim lim
3 13x x x
xx x
f xx x
x x2
1 1lim lim
3 1 3x x
xf x
x x
lim lim 1 1x x
f x x x
2lim 0
1 1x x x
1 1lim
1 1x
x x
x x
1 1 1 1lim
1 1x
x x x x
x x
2lim lim 1x x
f x x x x2lim 1x
x xlimx
x
2
11
lim1 1
1 1x
xx
xx x
2 2
2
1lim
1x
x x x
x x x
2
11
1lim
21 11 1
x
x
x x
2 2
2
1 1lim lim
1x x
x x x x x xf x
x x x
0;2D
lim 0x
f xlim 0x
f x0
limx
f x0
limx
f x2
limx
f x
2limx
f x
0D
limx
f xlimx
f x0
limx
f x0
limx
f x
1;1Dlimx
f xlimx
f x1
limx
f x1
limx
f x
1limx
f x1
limx
f x
3
3 4lim
3x
xx
limX
X3
3 4lim
3x
xx
2lim 1x
x xlimX
X2lim 1x
x x
3lim 2 3x
x xlimX
X3lim 2 3x
x x
2;224 0x2
2lim 4 0x
x2
2lim 4 0x
x0
lim 0X
X
22
3lim
4x x22
3lim
4x x
2
4 1lim lim 0
3x x
xx x
0lim cos 1X
X2
4lim cos 1
3x
xx
1lim lim
2 2 2x x
x xx x
2
lim cos 0X
X1
lim cos 02x
xx
1lim
2 2xx
2
lim sin 1X
X1
lim sin 12x
x
21
1lim
1x x21
1lim sin
2 1xx
x
0
sinlim 1x
xx0
sinlimx
xx
lim cos 1X
X0
sinlim cos 1x
xx
2
1lim 0
1x xlim 3 0x
f xlim 3x
f x
3lim 2x
x32f x xlimx
f x
41lim
2xx x41
2f x x xlim
xf x
1 cos 1x2 2cos 2x1 3 2cos 5x
x1x1 3 2cos 5x1 1
15 3 2cos x
1 11
5 3 2cosx x
xx
1lim lim 1
5x x
xx
1lim
3 2cosx
xx
2 23sin 3 3sin 3 3 1 sinx x x x x1 sin 1x1 sin 1x
0 1 sin 2x2 23sin 3 0x x x2 23sin 3x x x
2lim 3x
x2 23sin 3x x x2lim 3sinx
x x
0x1 sin 1x2 2 sin 2x x x x
2 2 22 2 sin 2x x x x x x x2 2lim 2 lim 2
x xx x x xlim
xf x
0x1 sin 1x2 2 sin 2x x x x
2 2 22 2 sin 2x x x x x x x2 2lim 2 lim 2
x xx x x xlim
xf x
1 sin 1x1 sin 1x x x x1 sin 1
2 1 2 1 2 1x x x xx x x
1 1 1lim lim
2 1 2 1 2x x
x xx x
1lim
2xf x
f2
2
2 1 ; 2
5 ; 2
f x x x x
f x x x x
2
2 2lim lim 2 1 1
x xf x x x2 1ff
2
2 2lim lim 5 1
x xf x x x2 1ff
f
f;22;
31
4f
10
4f
1 32 8
f5
14
f
11;
21
;02
0;1
f3;02;0 2;
0f x3;0
f0;22;40 2;4
0f x0;2
0f x0x1x03 0x10 2x
43
limx
f xlimx
f x
fx2' 3 6f x x x
' 0f x0x2x' 0f x0x2x
' 0f x0 2x
x
f x
1;00;12;3
:h x f x g x7 3
;8 4
2a3b1c1d
1a1b3c1d
2
3 11
1 1f x x
x x2
3 1lim 1 lim 0
1 1x xf x x
x x
2
3 1lim 1 lim 0
1 1x xf x x
x x
Cf1y x
1f x x2 2
3 1 3 21
1 1 1
xf x x
x x x
1 0f x x23
x1 0f x x23
x
1 0f x x23
x
C2
;3
C; 12
1;3
limx
f xlim 2 0x
f x x
: 2y xC
limx
f x
22 2
4 5 4 51 1
lim lim 1x x
x xx x x x
x x
2 4 5lim limx x
f x x xx x
2
2
54
4 5lim lim 2
4 54 5 1 1x x
xx x
x x x xx x
2lim lim 4 5x x
f x x x x x
C'2y x
limx
f xlimx
g x
1lim 0
2xf x x
1 3lim
2 2xg x x
fC12
y xgC
1 3lim
2 2xg x x
1 3lim 0
2 2xg x xlim 2 0
xg x x
'2y xgC
2
12 3
xf x
x x3;1fD
2 2
1 1lim lim lim lim 0
2 3x x x x
x xf x
x x x xlim 0x
f x
21 1
1lim lim
2 3x x
xf x
x x1lim 1 2
xx2
1lim 2 3 0
xx x
1lim
xf x
3lim
xf x
3lim
xf x
32 8x
f xxfD
3 322 8
lim lim lim limx x x x
x xf x x
x xlimx
f x
2
0
2 2 2 2 2 4limx
x x x
x
3
0 0
2 8lim limx x
xf x
x2
2
0 0
6 12lim lim 6 12 12x x
x x xx x
x
xsin 3g x x2cos 1h x xg x
f xh x
sin3 sin 33 3
3 3
2cos 2cos3 3
3 3
x g x g
x x
x h x h
x x
sin 3 0
sin 3 32cos 12cos 1
3
x
xxf x
xx
x
3
'3lim
'3
x
gf x
hgh
3' 0
3h
' 3cos3g x x' 2sinh x x
' 33
g' 33
h3
lim 3x
f x
0limsin 2 0x
x0
lim 1 cos 0x
x00
0
sin 2 1 coslim
sinx
x xx0 0
sin 2 sin 2 1 coslim lim
1 cos 1 cos 1 cosx x
x x x
x x x
0x0
sin 21 cos
2lim 2 2 2sinx
xx
xx
x
10
sin 2 1 coslim
sinx
x xl
x
2
2
lim 2 tanx
l x x0
2X x
2x X
2x0X
0 0
2 2lim lim 2
tantanX X
XXX
X
0lim 2 tan
2XX X2
2
lim 2 tanx
l x x
fDx2 1 0x x
2 21 2 1x x x x x2
2 2 2
112 2
1 1 1
x xx x
x x x x x x
x2 1 0x x2
12 0
1x
x x2
12
1x
x x
1 sin 1x0 1 sin 2x
0 1 sin 2x x x0x
2
12
1x
x x2
1 sin2 1 sin
1
x xx x x
x x2
2
1 sin4
1
x xx
x x24f x x
2lim 4x
xlimx
f x
/
ff x x
0 0
0lim lim 1
0x x
f x f xx x0 0
0lim lim 1
0x x
f x f xx x
f0
A2T3
' 22
f
2 3fT3
2 32
y x
f4 3 2' 5 4 9 2 4f x x x x x
26 2 4'
4x x
f x
2 3 2' 6 6f x mx m x m3 2 22 9 2 1f m x m x mx
cosf x x x xD' 1 cos sinf x x x x
sin cosf x x xD2 2' cos sin cos 2f x x x x
sin xf x
xD2
cos sin'
x x xf x
x
42 27 7f x x x3 2' 8 27 2 3 4 6 9f x x x x x
x24 6 9 0x x32
x' 0f xf
3;
232
x' 0f xf3
;2
cosf x x x' 1 sinf x xx1 sin x' 0f x
f
1 1f x
x xf2
1 2'
xf x
x x
' 0f xf
ff
2;2
'f
A
B
C
A
B
C
22' 3 2 2 2 3f x x x x32' 4 4 1 2 1g x x x x42 3' 5 3 1 1h t t t t
92
16'
3
ut u
u
2 1y x x
21 14 4
y x x
1
g2: 1u x x x:f x u
1 1f1 1g1
' 12
f1
' 12
g
f1
1 12
y xg
03
1 1 3x x3 3 21 3 3 1x x x xx03x23x
31x x01 3y x
ff
5;2fD
1' 0
2f' 3 0f
0 4' 2 2
2 0f
A1yB94
yC2 2y x
fCf
f0;2
1;0;M x y1M0y2 21 1x y0y
21 1y x
21 1f x x
0
0lim
x
f x f
x
f3 23 3 3f x x x x
fTA0
3 3y x
5;
2x
x23 3 3f x x x x
3;x3 3 0f x x; 3x
3 3 0f x x
/
xx e
2xx e
y ay b
2 22 2
2 22 2
11 1
1 11
x xx x
x xx x
e ee e
e ee e
22 2
1 1 1xx x
x x x
ee e
e e e2 2 2( ) ( ) 2 ( )x x x x x xe e e e e e
2 2 2( ) 2x x x xe e e e4
2 22 2
1 1( ) 2 2
xx x x
x x
ee e e
e e2
2
1
1
x x xx x x
x x xx x x
e e ee e e
e e ee e ekxx e
f x y f x f y
0x y0 0 0f f f2
0 0 0f f
0 1 0 0f f0 1ff
xf x f x f x x
0f x f x f1f x f x
x2 2 2 2x x x x
f f f f x
f
f0;
11
2xf x x e
1'
2xf x ef0;
limx
f x
1lim 1 lim 0
2x
x xf x x e
D1
12
y x
CD
( ) ( ) 1 sinxg x f x e x
sin 1x0;2
x2;2
A e
'( ) sin cosxf x e x x'( ) xg x e
2' '2 2
f g eA
3 22 3 11 6P x x x x
2 1 2 3P x x x x
0P x2x3x1
2 32 ; ;x e e e
1ln ;ln 3
2x
2 2 16
ln ln 3
x y
xy
8; 24 , 2;6S
2 2 169
ln ln ln 60
x y
x y5;12 , 12;5S
22 5 2 0t t1
;22
t
ln 2;ln 2 , ln 2; ln 2S
23ln 2 3f x x x x23 2 3
' 2 32 2
x xf x x
x x' 0f x22 3 0x x
' 0f x1x32
x
Cf32
1234log 2 1234 log 21234log 2 371E
log 371E n371 log 372n371 372log10 log log10n371 37210 10n
n372
3( ) xf x e2( ) xf x e52( )
xf x e8( ) xf x e
12( )
xf x e
fln 4 1f12( ) 2
xf x e
1y( )C
, , 2, 3,1a b c
lim ( ) 1x
f x( )C1y
lim ( )x
f x
'( ) 4 3x xf x e e
3ln
4x
'( )f x
18
f
( )Cln 2
( )Cy x( )
limx
f xx
21xe
11 1
1
x
x
ef x x x x
e1 2
1 11 1
x x
x x
e ef x x
e elim ( )
xf xlim ( )
xf x
lim ( ) 1 0x
f x xlim ( ) 1 0x
f x x
12C
C2C1
1 1( ) ( )
1 1
x x
x x
e ef x x x f x
e e
2 2
2 2 1'( ) 1
1 1
x x
x x
e ef x
e e
x
'( )f x
f
f
f0x
1x22,2 2 0,2
'( ) 2 2,21 1
x xf x x
x x'( )f x2 0,2x x
117
'( ) 0f x1;0 0,1;x'( ) 0f x0;0,1x'( ) 0f x0;0,1x
0,1x
'( )f x
0,0003f
1lim ( )x
f xlim ( )x
f x
(0) 0f0,1;f(0,1);f
(0,1) 0f0x0( ) 0f x
( ) 0f x0x
0,0018 0,00111y
(0,15) 0f(0,16) 0f0,15 0,162100,16
x1 cos 4 1x0xex xe f x e
lim ( ) 0x
f x
C2;2
k
kM k e
2 2 21
n
n nu e e e unu2e
nu20 1e0 1unu
x0;
' cos 4 4sin 4xf x e x x
' xg x e2
x kcos 4 1xsin 4 0x
2' '2 2
kf k g k e
C
2'2
f e110T
20, 2
1( )x xf x f x
e ef
xx exx xx xe e
lim ( ) 0x
f x
2'( )x x
x x
e ef x
e e0xx xe e'( ) 0f x
x
'( )f x12f
0x0 x xe e0 2x x x xe e e e1 1 1
2 x x x xe e e exh x f x g x
12
/
xx anx x
ln xxenx
1nn1n
n
x x
3 1 3 12 32 4 2 49 27 3 3a
3 3 1533 4 4 43 3 3 3a55 6534 123 81 3b
1 714 3 333 27 3c
12 3xln12 ln3x
e eln12 ln3xe e
ln12 ln 3x
ln 3ln12
x
1 14 8
xln 8ln 4
x
13
2
xln 3ln 2
x
15 2x xln 5ln 5 ln 2
x
2 13 4x xln 43
ln16
x
1 3 15
125x4
3x
25 5x xln 5 2 ln 5x x0;x
2 12 1 3
x
x
2.2 10
3 2 1
x
x12 1x; 1x
12
2
x
ln 2 ln 2x1
12
x2;x
xx anx x
2 3 5x x x2 31
5 5
x x
x x
2 3 5x x x2 31
5 5
x x
2 35 5
x x
f x2 3
ln ln5 52 3
' ln ln5 5
x xf x e e
2 2 3 3' ln ln
5 5 5 5
x x
f xlim ( )x
f xlim ( ) 0x
f x
x
'( )f x
f
2lim lim 1 x
x xf x x e
22 2
1lim lim 1
x
x x
ef x x
x x
2 2 2lim lim limx x
x x x
e x e xf x
x x x
lim ( ) lim 2 0x x
x xf x xe e
22lim ( ) lim lim 0x
xx x x
xg x xe
e
2
3lim lim 0
x
x xf x
x
20 0
3lim lim
x
x xf x
x2ln3ln3 ln3
2
2 22 2 2
ln 33ln 3
ln 3 ln 3
xx x xee ef x
x x x x
ln32
2lim lim ln 3ln 3
x
x x
ef x
xln 3X x
lim ( ) 0x
f x
'( ) (1 ) xf x x e
x
'( )f x1ef
y mf x m
(0,3573) 0,2499f(0,3574) 0,25001f
( ) 0f x0x1
( )f xe
1x
0u1nu
n nu u e
0u00nu0nunu e1 0nu0nu
1 1nun n nu u u e
0nu1nue1 0n nu unu
nu
e0
lnn nw u1nu
n nu u e1ln ln ln lnnun n n nu u e u u
1n n nw w u1n n nu w w
0 1nw w0 1 1 2 1 1( ) ( ) ( ) ( )n n n nw w w w w w w w0 1 .....n nS u u u
nunwnS
1
1( )
4u f
1( )
4f0vn nu v
D(0;1)J( 1;0)K1y x
lim 0x
xlim 0x
f x mx p
y mx pD1m p
J
1 ( )f x x x1 ( )f x x x
( ) 2 ( ) ( )f x f x x x2f x f x( ) ( ) 0x x
( ) ( )x x
2f x f x' ' 0f x f x' 'f x f x'f2
( ) xx ax b e2
( ) xx ax b e
ax b ax b0b2
1 ( ) 1 xf x x x x axe22' 1 '( ) 1 1 2 xf x x a x e
TJ'(0) 1f e
'(0) 1f a1 1e aa e2 2
1 1x xf x x axe x exe
x
xf x
e x'( ) 1xg x e'( )g x0x0x
g0x0x(0) 0g( ) 0g x
( ) 0g x1xe x0xe x
lim ( ) 0x
f xlim ( ) 1x
f x
C1yC
0y
2
(1 )'( )
x
x
x ef x
e x
'( )f x(1 )x
x
'( )f x1
1ef
Ty x
T( )
x
xg xf x x
e x( ) 0g x0xe xf x x( )x
;0 T0; T
' ( 1)(1 ) xf x x x e
' 0f x1 1x
' 0f x1x1x
' 0f x1x1x
f1;1; 11;
2lim ( ) lim x
x xf x x e
22
2
1lim ( ) lim lim lim 0x
x xx x x x
xf x x e
e ex
C
0y
0k
0k1x
40 k
e4
ke
1x
4k
e
1x4
( )f xe
( ) 2f x2f x
1;
1xf0;
0;x2f x
( 2) 7,39f( 1) 0f
0 2 7,392 1
2f2( 1) 2e2( 1) 2e
1 2e1 2e
121 2e2 ln
'( )2
xf x
x x
0lim ( )x
f xlim ( ) 0x
f x
2ex
'( )f x2ef
(1) 0f'(1) 1fT1y x
1g x x f x
1' 1 ' ln 2 1
2g x f x x x x
x x
'(1) 0g'g xln 2 1x x x
0;1ln 0x1 0x x' 0g x
1;ln 0x1 0x x' 0g x
1 0g
x
'( )g x
g
x0;0g x
x0;CT
1( )
1
x
x
ef x
xe'( ) 1xh x e x
x
'( )h x
11
eh
x1
( ) 1h xe
( ) 0h xx
( )g x( )h x
lim ( )x
g x2
lim ( ) lim 1x
x x
eg x x
x x
'( ) 1 xg x e
x
'( )g x
g
; ;x( ) 0g x;x( ) 0g x
f
lim ( ) 1x
f xlim ( ) 0x
f x
2
1 1'( )
1
x x x x x
x
e xe xe e ef x
xe
21
x
x
e g x
xe
2
2 2
22'( )
1 1
x xx x x
x x
e e xe e xef x
xe xe
x
'( )f x
( )f
( )f
f
( ) 0g2e
2
1 2 1 1 1( )
1 2 1 11
ef
e
( )f1,14 1,152,14 1 2,15
1 1 12,15 1 2,14
0,465 ( ) 0, 467f32 10
Ty x
211
x x x x
x
e x e x xe xexe
211
x x
x
e x e xxe
1( )
1
x
x
ef x x x
xe
1 1
1
x x
x
e xe x
xe
1 1( )
1
x x x x
x
x e xe e xef x x
xe
'( ) xu x xe
'( )u xx
x
'( )u x
u
( )u xx( ) 0u x
( )f x x1x
( )CT; 10;( )CT1;0
/
u xx u x e
u xx
u x
'F x f x
fHfH
gKhF
kG
3 32 2 2 212 2 2
2x x x xf x e e e e
f421
: 28
xF x e cc
2 2
1 1 1(ln 2) (ln 2)
f xx x x x
f1;1
:ln 2
F x cx
c
22
1 1
x x
x x
e ef x
e e0;I
f1;: 4 1xF x e cc
2 2
6 3 2 13
1 1x x
f xx x x x
f2: 3ln 1F x x x cc1 1
2 2
3 13x xf x e e
x x
f0;1
: 3 xF x e cc2 2
3 3
1 1 1 32 1 2 3 1
x xf x
x x
f1;31: ln 1
6F x x cc
sin cosf x x x
f21
: sin2
F x x21
: cos2
G x x
3 22 13 2
y x x x c2 1y x x c
x2
2 2
1 1' 1
xy
x x1
y x cx
32
2y cs x cc
1
31
27
2 4sin cos cosf x x a x b x
2sin sin cos 2f x x x x2 2sin 1 cos cosf x x x x
2 4sin cos cosf x x x x
3 51 1cos cos
3 5F x x x
4 2 2
6
cos 3sin cos'
cosx x x
u xx
2 2 2
4 4 2
cos 3 1 cos sin 3 2'
cos cos cos
x x xu x
x x x
2
1 2'
3 cosv x u x
x
v0;4
12 tan
3x u x x kk
0 0V0k3
1 sin2 tan
3 cosx
V x xx
3 3
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