Www.science-ki.blogspot.com Math PC (18)

demahoM iluoatsuoM rf.aliov.etis.shtambara//:ptth

-I x0 l - 1 0 0 -

x gx3 ) ( =x x f2 ) ( g f

x =

f ( -1 ; ; f; 0 ; / 0 0[ ] ) (} { [ ] ( (

g ( -2 -2 ; ; g; 0 ; / 0 0[ ] ) (} { [ ] ( (

-3 x g) ( x f) ( x

012 015

01001 ////////////////////// ///////////////////// 0

01001 015 012

: 0 x f) ( 0 x 0 x f) ( 0 x

0 x 0 f ) (

00 milx

= x f g

0 f

0 x 0 f ; ; x f x D xf0 0 0) (

) ( 0

0 milx

= x f

) ( ) ( * 0 0

0 mil 0 milx x

= = x f x f) ( 0 g f *

00 milx

= x g x0 mil0) ( = x f


` \ n a*

0 0n0 mil 0 milx x = = x a xa

) ( x u x f I x) ( ) ( 0 I

00 milx) ( = x u

00 milx

= x f = I0 ;} { [ ] ; 0 x u x f x0 ;) ( ) ( } { [ ]

) ( 0

0 milx ; ; x u x D xf0 0 0) ( = x u

f0 0 0) ( ) () ( = ; fni) ( x u

x D xx u x f

; ;

; ; x f x D xf0 0 0) ( ) (

00 milx

= x f

0 0

0 nis mil ; 0 mil1x x1x x x x

+= = x0 l -

x0 f x0 x l x f) (:

0 h x0 x 0 l x f) ( =x x h0 0 +l h x f0) (

x0 f

0 + l h x f h0) ( x0 x l f 0 h

) ( 0

milx x

= l x f

) ( ) ( * 0

00mil milh x x

= + = l h x f l x f) ( ) ( *

0 00 mil milx x x x

= = l x f l x f x0 * ) ( ) ( *

0 00 mil milx x x x

= = l x f l x f) ( *

0

*0 mil0n

x x ` \= x x a n a

4

9 mil1 2x3

xx =


) ( x u l x f I x) ( ) ( x0 I

0

0 milx x

= x u) (

0milx x

= l x f

2 0

2 soc 2 mil1x

xx

= + ) (

0milx x

= l x f x xmil0) ( = l x f -2 -

x0 f x0 f

) ( ) ( 0

x xmil0 = x f x f

\ ` a n*) ( 0 xa xn

0 x x

f x0 x0 f

x0

) ( \ f ) (

1 1 21

2 1

x x fxx

f

= =

1 f -

\

\ x0 P = x Q x x x P x P0 0) ( ) ( ) ( ) ( Q

+ + + + a x a x a x a x Qn nn n.......0 1 11 ) ( + + + + =a x a x a x a x Qn nn n.......0 1 11 ) ( + + + + x x x M x Qn n)1 ....... (1 ) ( n i;...1;0} { ai M

x + = x x1 ; 1 pus0 0) ( + x x x1 10 0 + + + + M x Qn n)1 .... (1 ) (

x x k x Q x x0 0) ( + + + + = M kn n)1 .... (1


x x k x P x P0 0) ( ) (

0x x0 mil0

= x P x P x xmil0 0) ( ) ( = x x k x0 P

- 2 2

1 22 4 3 mil ; 2 7 milx x

+ x x x x2 2 3

1 5

mil ; mil2 5 3 5 4 4x x1 5

x x x x x x x

+

-

:1 2

1x fxx 1

2 ) ( 1 1 1

2 1 mil mil mil1x x x1

x x fx x = + = =

) ( g ) (

1 1 21

2 1

x x gxx

g

= = 1 \1} { f

1 f g

x0 l x0 f 0) () ( ) ( g

D x x f x gfl x g

= = f x0

x0

x0 f

2 ) (

0

8 2 32

2

x fx xx

x

= + = +

2 ) (0

nis1

0

x x fx

x

= =

- - 32 1) ( ) ( f

1x x

x fx

=+ (Cf ) \ = Df1} { + =x x g2 ) ( g +;1[ ] f *

) ( 1 1

3 2 mil milx x = + = x x g

1 3 f ) (

13 milx

+x f) ( =

11

3 milxx

= x f;

=x x h2 ) ( h 1;[ ] f * ) (

1 13 2 mil milx x = = x x h

1 -3 f


) ( 1

3 milx

x f) ( =

11

3 milxx

= x f

; 0 + x x;0 0[ ] f

; a0 +a x x;0 0[ ] x0 l f

) ( x0 l x0 0

milx x

+l x f =

) ( 0

0

milx xx x

= l x f;

) ( ) ( ) (

0 0 0

mil mil milx x x x x x

+l x f x f l x f = = =

x0 f

) ( 3

0

2

0

x xx f

xx

= = +

) () (

2

2

0

2 4 42 2 2

2

x x x x fx x x x f

x

= + = + + =

;

) ( ) ( x0 f

0mil0

x x+x f x f

=) ( ) ( x0 f

0mil0

x xx f x f

= x0 x0 f x0 f

) () ( 1 f a -1

1 31 1

x xa x x fx x x f

+ = + = ;

x0 f -2

) (2 0) (

2 1 2; 2

2 1

x x x fx

x x x f

= =+ = ;

) (2 0) (

0 nis1; 0

0

x x x fxx

x x x x f

= ==

;

4- b a;] [ f b a;[ ] b a;[ ] f

a b a;[ ] b a;] [ f b


b a;[ [ b a;] ]

b a;] [ b f b;) () ( a f a;) () (

+ -II + 0 - 1

x f1 ) ( f x

= Cf -1 -2

0101001

010121 01019

x 01001

x f) ( -3

; ; + B f B; ; 0 0[ ] ) ([ ]

3- ; 0

; x f B x) (; B0 x x f x01 1 ) (

x ; ;

= B1 ; ; ; x f B x B0 0) ( + x 0 x f) (

0 mil) ( x

= +x f + a;[ ] f

+ x 0 x f) ( 0 mil) ( ; ; ; x f B x B0 0) (

x = +x f


1x xn0 mil ; 0 mil ;* ) (

n kk k+ +x x

` \= =

2 + a;[ ]

0 mil) ( + x u x f a x;) ( ) ( [ ]x

= +x f x0 mil) ( = +x u

2 mil7

+ +x x3 4

2 2 ; +x x3 42 2 7 7

x x3 42 +

0 mil7+x x

2 =0 mil7

+x x3 4 +=

+ l - 2 + a;[ ] f

+ x l x f) ( ; ; ; l x f B x B0 0) ( mil) (

x = +l x f

2

21 mil2x1

x+x +=

l -3 a;[ ] f

mil) ( x l x f) ( x

= +l x fmil) (

x = l x f

mil mil) ( ) ( f -

x x =+ x f x f

mil mil) ( ) ( f -x x

=+ x f x f -III

1


x0 I f ) (

0milx x

l0 I f = l x f

2 x0 f ) (

0

milx x

; l x f J x0 ) ( x0 J l0 = l x f 3

x0 I g f) (

0milx x

l l' I g f = l x g x x' mil0) ( = l x f 4

x0 I h g f) ( ) (

0 0mil milx x x x

= l x h x xmil0) ( I g h f = = l x g x f -VI x0 g f

x0 f g f f +g f ) ( ) ( ) () (

0 0 0mil mil milx x x x x x

= x g x f x g fx x x x x xmil mil mil0 0 0) ( ) ( ) () ( + = + x g x f x g f) ( ) () (

0 0mil milx x x x

= x f x fx x x xmil mil0 0) ( ) ( = x f x f) (

00 milx x

= x f x fx x x xmil mil0 0) ( ) ( x f

) ( 0

0 milx x

) ( x g) (00) (

0

milmil

milx x

x xx x

x fx fx g g

=

= x0 + = x0 - V

x0 x0 -* x0 x0 -* x0 x0 -* g1 x g0 0) ( x0 g f -*

fg

x0 x0 f x0 x0 f -* x0 + b xa f x) ( f -*

x0 + b xa f x) (

http://arabmaths.site.voila.fr Moustaouli Mohamed

1- 2 2

2 21 3

3 5 2lim ; lim1 6x x

x x x xx x x + +

2

2 1

3 2lim ; lim 5 22x x

x x x xx

+ 2- ( ) 2 22 3 1x xf x x x

+= ( )2 1 2g x x x= + ( )( )

2

2

2 1

2 1

h x x x x

h x x x

= = + ;

( ) 22 3x xt xx=

VI - ; sin tan

2 2x x x x

1- sinx x cosx x tanx x * ; sin

2 2x x x 0lim 0x x = 0lim sin 0x x =

sinx x 0 sin( )x ax b + 0 * 2

0 0lim cos lim1 2sin 1

2x xxx = =

cosx x 0

0 0

sinlim tan lim 0cosx xxxx

= = tanx x 0

* 0x \ 0

00

0 00

0 0 0

lim sin lim sin( )

lim[sin cosh sinh cos ]

sin cos0 sin 0cos sin

x x h

h

x x h

x x

x x x

= += += + =

sinx x 0x

sinx x cosx x \ tanx x /

2k k + \ ]

sin( )x ax b + cos( )x ax b + \

tan( )x ax b + 2 -

0

sinlimx

xx

; sin tan2 2

x x x x 1 1 1

tan sinx x x 0x

sin sin sintan sinx x xx x x sincos 1xx

x

0

lim cos 1x

x = x sin x 0 0sinlim 1

x

xx

= *


22

2 20 00

2sin sin1 cos 1 12 2lim lim lim2 2

2x x

x

x xx

xx x

= = =

* 0 0

tan sin 1lim lim 1cosx x

x xx x x

= =

0

sinlim 1x

xx

= 0

tanlim 1x

xx

= 201 cos 1lim

2xx

x =

0

sinlim 1x

axax

= 0

tanlim 1x

axax

=

0

2

sin3 cos 2lim ; lim3 1 sinx x

x xx x +

0

sin 3lim4xxx

2

20

sinlim3xx

x

0

sinlimsin 3x

xx

0

tan 3limsin 2x

xx

0

1 cos 2limx

xx

4

cos sinlim

4x

x x

x

0

cos 4 cos 2limsin 4 sin 2x

x xx x+

VII- 1 - + 0x

( ) 1f xx

= 1- fC

2- 1001010

121010 91010 10010 x

( )f x ) (

* f 0.

( ) ( ) ( ) ( ) ( )0

lim 0 0 0fxf x A x D x f x A = + ; ; ;

f 0x. ( ) ( )

000

lim limx x h

f x f x h = + + = +

f 0x. ( ) ( )0 0

lim limx x x x

f x f x = = +

* *;n k + ` \ 0 0

lim ; limnx x

k kxx

= + = +

2 - + +


.+ a;[ ] f * ; ; ; ; + = +A x f B x D x B A x f xf 0 0 mil) ( ) ( ) ( ) (

mil mil) ( ) ( x x

+ = =+ +x f x fmil mil) ( ) (

x x =+ x f x f

) ( x u x f) ( ) ( , Ix *

0

milx x

+ = x f x xmil0) ( + = x u) ( x u x f) ( ) ( , Ix *

0

milx x

= x f x xmil0) ( = x u x0 x0 +

; x x; 00 0[ ] ) ( + x x;0 0[ ] a;[ ] + a;[ ] I -IIIV . g f

: + x0 x0 x0

- +g f g f

+ + l0 l l0 l

+ + + +

- g f g f

l + l0 l l l0 l

+ 0 0 + + + + +

: f \ f

f x


- f g f

g

0 + l 0 l

l +0 l0 l

l 0 l0 l

0 0 + + + l l0 l + l l0 l

f - f f + +

-XI

-1n mil `n*

x + = +x

n mil n -x

+ = xn mil n -

x = x

= x xn0 mil1 `n*

- 2) () (

10 1 1

10 11

.......

....... 1

n nn n

n nn n n

n n n

a x a x a x a x f

x a x faa ax a x a x a

+ + + + = + + + + =

n n n xn n n1 ....... 1 mil1 10 1 aa ax a x a x a

+

= + + + + =+ +x a x fx xnn mil mil) (

+ x


5 2 5

7 3 7

lim 4 3 5 1 lim 4

lim 3 7 31 lim 3x x

x x

x x x x

x x x x+ +

+ + = = + + = = +

3 -

x +

5 2 53

2 2

7 3 7

9 2 9 2

4 3 5 1 4 4lim lim lim33 1 3

3 7 31 3 3lim lim lim 03 4

x x x

x x x

x x x x xx x x

x x x xx x x x

+ + +

+ + = = = + + + = = =+

5 2 5

5 4 57 3 5 1 7 7lim lim

33 1 3x xx x x xx x x+ ++ + = = +

2

2 21 1

2

1 2

2 2 31 0

22

2

20 1

2 1lim ; lim2 3 2 3

5 2 6lim lim1 2

3 2 1 1lim lim3 2

2 3lim 2 lim2 1

4 2 2 2lim ; lim1

3 4 2lim ; lim1 4

x x

x x

x x

x x

x x

x x

x x xx x x xx x xx xx

x x x x

x xx xx

x xxx x

x xx x

+

+

+ +

+ +

+ ++ + + +

+ + +

+

+

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Www.science-ki.blogspot.com Math PC (18)