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Maths

350

1lim 0

x x

1lim 0

x x

0

1limx x

0

1limx x

limx

x

1lim 0

x x

1lim

x a x a

1limx a a x

nx

n

lim n

xx

lim n

xx

n

lim n

xx

n

1lim 0nx x

00

0

0

0

0

0 0

x

x

x0x

x0x

uvff v u abc

limaxu x b

lim

x bcv x

lim

axcf x

g x f x h x limax

g x

limaxh x

lim

axf x

f x g x limax

g x

limax

f x

f x g x limax

g x

limax

f x

f x g x lim 0ax

g x

limax

f x

0

sinlim 1x

xx

0

tanlim 1x

xx

0

sin limsin x

xx

0

1 coslim 0x

xx

20

1 cos 1lim2x

xx

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Maths

351

f fC ; ,O i j

0

limx x

f x

0

limx x

f x

fC

0x x

0limx

f x y

0limx

f x y

fC

0y y

lim 0x

f x a x b

lim 0x

f x a x b

fC

y a x b

limx

f x

y a x b ( )

limx

f xx

0( )

limx

af x

ax

( )

limx

f xx

( )

lim 0x

f xx

limx

f x a x

limx

f x a x

limx

f x a x b

limx

f x

y a x b ( )

limx

f xx

fC x x fC y y

fC

y a x b fC

y a x

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Maths

352

fIa

fa limx a

f x f a

fa 1limx a

f x f a

1 1 2

fa fa 2limx a

f x f a

2

fI fI

I f I f I

;a b ;f a f b ;f b f a ;a b ; lim

x bf a f x

lim ;x b

f x f a

;a b lim ;x a

f x f b

; limx a

f b f x

;a b lim ; limx a x b

f x f x

lim ; limx b x a

f x f x

f;a b k f a f b

f x k;a b

f;a b k

f a f b f x k;a b

f;a b 0f a f b

ab 0f

f;a b 0f a f b

0f x ;a b

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Maths

353

ffDafD

fa limx a

f x f af a

x a

fa lim dx a

f x f af a

x a

d gf a f a

fa fa lim g

x a

f x f af a

x a

u v a u u v 1v

uv

u nu u v

u v a u

u v v u

2vv

2

u v v uv

2u

u

1 nn u u 1n .v u v

f x f x

0 a a 1 x a a x

1. nn x nx 1n

2

1x 1

x

1nn

x 1

nx 1n

0; 1

2 x x

xsin cos x cos x xsin sin a a x b cos a x b cos a a x b sin a x b

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Maths

354

0

0

0lim

x x

f x f xa

x x

f0x

0f x a

fC 0 0;A x f x

0 0 0y f x x x f x

0

0

0lim 0

x x

f x f xx x

f0x

0 0f x

fC 0 0;A x f x

0y f x

0

0

0lim

x x

f x f xa

x x

f

0x 0df x a

fC 0 0;A x f x

0 0 0dy f x x x f x

0

0

0lim

x x

f x f xb

x x

f

0x 0gf x b

fC 0 0;A x f x

0 0 0gy f x x x f x

0

0

0

0

0

0

lim

lim

x x

x x

f x f xx x

f x f xx x

f

0x

0 0g df x f x

fC 0 0;A x f x

A

0

0

0lim

x x

f x f xx x

f

0x

fC 0 0;A x f x

0x x

0

0

0lim

x x

f x f xx x

f

0x

fC 0 0;A x f x

0x x

0

0

0lim

x x

f x f xx x

f

0x

fC 0 0;A x f x

0x x

0

0

0lim

x x

f x f xx x

f

0x

fC 0 0;A x f x

0x x

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Maths

355

fCf y a x b

f x y

0f x y fC

0f x y fC

0f x y fC

f

fx Dfx D

f x f x

fC

f

fx Dfx D

f x f x

fCO

; fC

fx D 2 fx D

2 2f x f x

x fC

fx D 2 fx D

2f x f x

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Maths

356

fC gCfg ; ,O i j

f x g x b fC gCb j

f x g x a fC gCa i

f x g x a b fC gC ;v a b

f x g x fC gC f x g x fC gC

f x g x fC gC

f x g x

0x f x g x fC gC

0x f x g x fC gC

f

f x g x 0g x f x g x fC gC

0g x f x g x fC gC

fCf y a x b

m

f x m fC

f x a x m fC

f x m x b fC 0;b

2f x m

f x m

fC

2y my m

f x f m fC

y f m

http://touamimaths.webnode.fr

(01) :

: :

01

1 xxxx

24 2lim41

21lim

xx

x

2 72lim 23

xxx

53

2lim

3

xxx

x

3 2lim 47

xxx

6113

lim6

5

xxxx

x

02

122

lim2

2

xxx

x4

)1)(4(23

lim2

4 xxxx

x

225

103lim

2

2

5

xxx

x522 )2(

212lim

xx

x

334

35lim

2

2

1

xxxx

x6

1510105146

lim2345

23

1

xxxxxxxx

x

03

1x

xx 2

3lim

24

132

lim31

xx

x

230lim

x

x

x 5

xx

xx

2

2

0

1lim

33

9lim

2

3

x

x

x 6

1232

lim2

3

xxx

x

04

13

3lim

3

x

xx

5642

2lim

24

x

xx

22

2lim

2

xx

x 6

12

lim2

xx

x

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3 3lim 2

xxx

724

11lim

0

x

xxx

4 124lim 2

xxxx

8xxxx

xxxx

22

22

2 12

1143lim

05

1x

xx 3sinlim

05

xx

x tan12sin2

lim4

2xx

x sin2sin

lim0

6

23

sin

1cos2lim

3

x

x

x

3x

xx

sinlim

07

xxxx

x cossin1cossin1

lim2

4x

xx sin

cos1lim

0

8)2sin()2tan(

5tan.lim

0 xxxx

x

06

11f 1ℝ 2)1(

x

xxf

2I1Ix 610xf

0712f

21

15

x

xxf

2 1;1x 1000xf 08

f 1416

xx

xf AAx 6,1;4,1xf

09

x

x

x

)1(42lim

1

cos3lim

xxx

x



10

4xx tan2cos2 xx

-2cos2

1tanlim

4

x

x

x

11f

1

)sin1(2

xx

xxxf

1f

2xxxx

21

12

3x 24xxf f

12f xxxxf 12 2 1xxx 212 2 xℝ+ )(lim xf

x

2)(lim xfx

13

f 1

sin.22

x

xxxf

1f

21x 1

22

x

xxf

3f

14 n

24lim 11

xxxn

x

243

lim9

48

xxxxx n

x

15

nm1

)1.....(.lim

21

1

m

nnn

x x

xxxn

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16

xx

x

x

x

x

xx 444

......21

lim

17

f 9235423

5522

2

xxx

xxxf

1fDf

2xfD

72354

134

223

335

2

22

1

2

2

xx

x

x

x

x

xxf

3 18977

lim1

xfx

18

fℝ

1;15,0

1;22

xxxf

xxxxf

fℝ

19

f 2

11.2

xxxf

1f 2f 3 12 ff10 x

20

f 143.1

21232

2

xxx

xxxf 12 f

f20 x 21

f

1;1

2

1;12

xx

xx

xxxxf

f10 x .



22

fℝ

0

0;42 2

f

xx

xxxf

fℝ

23

;0f

2;1

2;

coscos

cos

xxf

xxx

xxf

f20x .

24

f

34

sin2

3

xxx

xf

1f 2f 3f

25

fCf 1f 2 xf ,0,4, 3f 4 fC 5 f 3;11 I ;22I 0;43 I 4;4 I

26

f*ℝ x

xxf 1

1h 2

11)1(

3311 2

hh

hhh

fhf

2f1 1f 3f1

2 3 4-1-2-3-4-5-6-7

2

3

-1

-2

-3

-4

-5

-6

0 1

1

x

y



27 fCfℝ 12 xxxf 1fℝ

20h hh

hhhf

211

3 hhf 11 h0

4 fC

28

f x

xxxf

1

)1(2

1f 2f01 3 xf

29fℝ 12 xxxf 1x xfxfx 12 2x 0.12 xfxfxxfx

30fℝ fC T0 1 )0(f )1(f T 2 fC T 3 3)( xf3)( xf0)( xf 0)( xf

30 f ;x 2cos3cos3 xxxf

( T )

(Cf )

2-1-2-3

234

-10 1

1

x

y

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31fℝ

1.3

2

2

xbxax

xfab

fC ab fC34 xy0

32

afℝ xxxaxf 33. 23 af1x

33fff ,,)(nffℕn

f 0 ℝ x

xf1

1 xf xf xf

2n 1

)( !)1(

n

nn

x

nxf

34

04335 xxx 1;0

35

0123 xx

21

;41

36

f 2;1 124 xxxf 1f 2 3xf 2;1 3210

37

xx

cos22

1

0;

2



381: xxf 3: xxg

fC gCfg0x

83

87

0 x

39101. xxx

21

41

201. xxx

401043 xxℝ310 2 43 xxℝ

41 Cxx cos

dxx23 0;I

1xx23

cos I

2I xxxf23

cos fI xf f

3xx23

cos I

42

AB H

O B

hkm /40 hkm /40

10AO4BO H

AB

(C)

(d)

0 1

1

x

y

A H

B

O10

حقلال4



43fD

1ff 2ff 3fTf T

44f ;1 x

xxf

11

1f 2 0xf 2;1 3210 4012 23

45

fx 12

572 2

xxx

xf

fC jiO

,, 1f fC 2 fC 3 fC3 4 fC

5 0f 1f

2

5f

6 fC

7 12

572 2

x

xxxg

g g

46fx

32152

2

2

xxxx

xf

fC jiO

,,

1 31

1

xxxf

2 f fC

g

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fC50 x 3 1: x fC

4f32

1522

2

xxxx

x

5 fC

6mmfx 3.

15.2

2

xmxxmx

xf

mmf 4;1 mf

47I f

2)1(4

.

x

bxaxf fC

ab fC 7;3

II f 2)1(

43

xxxf

1f 2 0xf0x34 0 x 3 fC fC 4 fC 5m 0725412 23 mxmxmx

48I g ;1 132 3 xxxg 1g 2 0xg 7,1;6,1 3x xg ;1

II f ;1 1

13

xxxf

fC jiO

,,cm4 1

xf

x 1lim

xfx lim

2x ;1 23 )1(

)(

x

xgxf

3f

4

11

32

f f110

5 fC0



6 x 1;1 1

)1(1

3

3

x

xxxxf

fC 7 fC

49fx

44

22

x

xxxf

fC jiO

,, 1 fDf f 2 xf xf 3f 4 2: xy 2: xy fC fC 5 fC T fC xx fC T6 fC0n1n0n1n 7082 23 xx6,15,1 8 T fC

50f 2 ℝ

)2(2633

xxx

xf

)( fCf),,( jiO

1ab2x 2

1 2

xb

xaxf

2 f f

3 2121 xy2x

PM )( fCx PMx PM0 )( fC



51

I )( fCf),,( jiO

0;

)1(1

2

0;21

123

2

2

xx

x

xxxx

xf

1 f f00 x 2f 3)( fC 4)( fC11 x2x10 2 x 5)( fC

II g 1cos2

1cos2cos3 2

xxx

xg

1g 2g 3 xgg

52

mfx 1.2

xxmx

xfmℝm mCmf jiO

,,

1 mC1m 2 mC

3ab,c 1

.

x

cbxaxfm1m

4 2;1 mm mC mm 1 ℝ 5 xfmmf mmf mmf

53 fx 33 2 xxf )( fC),,( jiO

1f 2f1 3f 4 )( fC



)( fC 5)( fC2 6)( fC

54 f xxxxf 41 2 )( fCf),,( jiO

1fDf 2 xf

x lim xf

x lim)( fC

32)(lim

xxfx)( fC

)( fC d32 xy 3f04 4f xf f 5)( fC

55 fx

1

12

xxxf

)( fCf),,( jiO

1f)( fC 2 0xf0x5,101,1 0 x 3)( fC

4g x

xxg

1

12)( gC

g )( gC)( fC )( gC)( fC 5)( fC2 xy2x3x

56 fgℝ 1

25

3 2 xxxf 125

3 2 xxxg

)( fC)( gC),,( jiO

1 xf xg 2fg 3)( fC)( gC 4 )()( gf CC



025.24114 22 yxxy 5u

jiu

23 yx;M),,( jiO

yx ;),,( uiO

xyxy ),,( uiO

57

f

;1;21

2

1;;11 2

xxx

x

xxxxf

)( fCf),,( jiO

1f 2f1x 3f)( fC 4)( fC 5 yxn ;43 x xfyx 1 6 )( fCf 1; 1;2

58

fℝ

0;0

0;112

x

xx

xxf

)( fCf),,( jiO

1f 2f00 x 3 xf

x lim

f11 x

1;1

11)(

10;1

11)(

22

2

22

2

xxx

xxf

xxx

xxf

f ;0 4)( fC



2 3-1

0 1

1

x

y

iJ M

A

59 mf 12. 22 xmxmxfmm

)( mC),,( jiO

1mmf 2)( mC

3mfm1

m1

431m)(

31C

5g 191

2.31 2 xxxg gC

gC)(31C xx gC

60

),,( jiO

M 0;x0x OMxf 1 f xxf

xf

xlim

2AMO ˆx x

1lim

xxf

x

3 cMOM HOA ˆ xxxf sin0 xxf

x

lim

4xy )( fCf

61

xy

10x

AM 1;1 A

xxM

1;

M AM

1 AM2AM x xd 2AMxd

20x 3

2x

xfxd f

3 f ;0I

f ( x ) = OM

0 1

1

x

y

i

J

M

A



0xfI210 xfI 4d 5MA AMM

62 1f

nAxAxAxxf

1

......11

21

12 ,,......, AAAn 12...... AAAn

201

......11

21

nAxAxAx

3 05

13

11

12

1

xxxx

63 1g 0; cos sing x x x x g g x 0;

2f 0; sin xf xx

0x 0 1f

f 0; 30f

x3

0 sin6xx x

0; 3

sin6xx x x

' x " x "' x f0 ' 0f 4Cf3cm

64

1g 0 0g 2 1sing x xx

0x

g0 Cg ; ;O i j

CO



2 1 0gk

k

1k 0;

k 3ACCAA

65 I f 4sin2 xxxf )( fCf),,( jiO

1f 2 0xf4,23,2 3)( fCcm2

II g xxg sin421

1 g

2x 21 xg

3 nuAu 0ℝ A n nn ugu 1

n nn uu21

1

n

02

1uu

n

n

nunx

u

lim

66

fℝ xxxf cos2cos21

)( fC),,( jiO

1 f2 )( fC 2 xf f x xxxf cos21.sin xf ;0x 3 f ;0 )( fC ; )( fCℝ

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67 Fℝ

11

2

xxF 00 F

F xF C),,( jiO

1Gℝ xFxFxG Gℝ xGℝ x 0GF

2H ;0I

xFxFxH

1

HI xH Ix Ix 12FxH 12lim FxF

x

C

3T

2;

2 xxFxT tan

xT T 1F 4Fℝ5 C1 01


lyceeroutedoran.123.fr…واضيع للمراجعة... · lyceeroutedoran.123.fr ... Maths 351

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Transcript of lyceeroutedoran.123.fr…واضيع للمراجعة... · lyceeroutedoran.123.fr ... Maths 351