# Full text of "Cours Math"

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JLLj] S^cla Jxxluija

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•lie JJ JjjJ Iglib La (j\£ bj JLijjJ ojcla JaxIuiJ j

Jlla

im r ^ +O0 x 2 + 5x - 7 lim^.ooX 2 + 5x - 7 lim^.™-^ — lim

X-> + CO A T JA / x->-00 A T JA / X-> + O0 2_ x _g X->-2 x 2_ x _ 6

3x-9 ,. 5x / -3 ,. x 3 -8 ,. x 3 -8

im x ^ 3 — hm x ^ +00 — hm x ^ +O0 — hm x ^ 2 —

x 6 x 2 -x-6 x + x 2 -x-6 x + x 2 -x-6 x z x 2 +x-6

X

5x 2 -3

X— > + 00 7 -

x / -x-6

li

lim^-^.oo Vx 2 -

- 1 + X

im x ^ +O0 Vx + 3 — x lim^-^.oo Vx 2 — 1 + x lim x ^_ O0 Vx 2 ~^T +

im x ^ +O0 Vx 3 — 1 — x lim x ^ +O0 Vx 2 — 5x + 1 — x lim x ^ +O0 Vx 2 — 1 — 5

X

im x ^ +O0 V2x 2 + x - Vx 3 + 1 lim x ^ +O0 Vx 2 + x - 1 - Vx 3 + 5x 2 + 3

Vx+l-2 ,. Vx+l-5+x ,. V4x+l-V5x 2 -ll ,. Vx 3 -1-V^-1
x A x 2 -x-6 x 6 x 2 -9 x z Vx 3 +8-2x x y Vx 2 -1

Ln(x)

im x ^ +O0 Ln(x 3 + 2x - 1) -Vx lim,^ Vx - 1 ■ Ln{x 3 - 1) lim^i

yx-i

im x ^ +O0 x 2 + 5x + 3 - e^ \\rr\ x ^_ m e x {x - x 2 ) lim x ^ +00 3x - Ln{e 2x + Vx)

im x ^ — : — - li m x ^ +O0 xLn( 1 + -) lim x ^ +O0 xLn(x + 1) —xLn(x)

xsin(x 2 + l) .. e x -cos{x) ■■ xstn(x)

IITI x ^ +0O - " m ^0 Ln(x+1) " m ^° C0 5(x)-l

im x ^ xLn(x)sin(-) lim x ^ n ° 5 2 lim x ^ +O0 sin(x)Ln(x)

\xj (sin{x))

]a , M JMI ^ JSVt J* 5U J£ /(x) = iial— »cii f{a)-f{b)<0 j^ bl

c = Sup(A) c£J j A= {i £ [a , I)] : /"(x) < 0} <cj^l jU>VI u^ iiL

/(c) = J^-^Wu^

/([a , &]) = [/(a) , /(£)] u^ [a , b] Jl^l ^ ^lji« j Sj-i-- *Jb / cjjIS bl

/([a , 6]) = [f{b) , fia)] d* [a , b] JMI ^ ^-SLsi. j Sj-l*. ^ib / cjj\£ bl

<-^ /(]a , M) = ]a , /?[ Oi* ]a , M JW-»1I Jfc sjjI jl«j Sj-lab« AJb / cjj\£ bl

fa = lim^ a /(x)
1)5 = \\m x ^ b f[x)

£***> /(]a , M) = ]/? , a[ u^ ]a , M JW-»1I ^Jc sjjI jwj Sj-i— AJb f cA£. bl

fa = lim^ a /(x)
I/? = lim^ b /(x)

/? ^*]| j lim x >^ a /U) ~ /(a) = /? G 5R c^i£ b] a u^j c> L3i£i^ ^Lii bi / *Jhli & J_£

a ij^ (Jt y 7 4JUI (jjA* t^-AJjj

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d Aic /" 4JUI (jjii-o cs -ojaiJ 4 jlfrjll lilii <Lajii

b* ^ ^i^ ilt (3lili!>U aIAI f aJU ciil£ bl I Q D f JW-a J& jlili!^ 4lii Uii / 4JIJI (> Jli

JU^ll

Jlla

V /\(i)=i j 1 -<*a\ *c \$&zyi mi f( x ) = y/x y^i(*

lim^-, - — = lim^-, -^— = -

x L x-l -Jx+l 2

lim^of = lim x ^ ^= +00

jV <-^l Xe. jlili!^ aIAI j^ f(x) = yfx ^ (*

jV /\ (a) = 5 j a G 5R *-*i c?i ^ jUiiSSU *L15 /( x ) = 5x + B *JUI (*

{Sx + B) -{Sa + B) Six -a)
I im — = lim = S

x->a x — a x^a x — a

uV fH ) = cosia) j a£5K *-*i c?i ^ li*^ *Ui /"(x) = sinix) *JUI (*

,. sin(x) -5in(a) ,. 2cos(— ^ )sin(— 2 J 5in(y) , . . .

lim = lim — y _ „ — - — - = lim cos(y + a) = cos(a)

x^a x — a x^a 2— — v^° y

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^Jj Ld (jAULLyaj AaLLoJI

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.AjJLill -Icljall LjjJ j (jUu*^U Alii AJb ^ aj-l*-a JJC- (jjjA* dlli j

[/(x)+^(x)]\ =/Ux)+^(x)
[/(x)-^(x)]\ =/Ux)^(x)+/(x)^(x)

/(*)

#(x)

* = fHx)g{x)-f{x)gHx)
g 2 {x)

[g[f[x)]) x = g x (y) ■ f x (x) avec y = fix)

f x = /(y)

[/-Mx)]^^ m, e c P-/.^

f'(y)

[f a (x)]\ = fHx) ■ f a ~Hx)

f x (x)

[Ln[f[x)W = [ 7 rr
f(x)

[efMf =fHx)-ef {x)

fHx) = f[x) ■ [Ln[f[xW

(sin[f(x)])\ = fHx) ■ cos[f(x)]

{cos[f(x)W = -fHx) ■ sin[f(x)]

(tg[f(x)W = j f , { *\„ 2 = fHx) ■ [1 + (tg[f(x)]) 2 ]
[cos[f(x)\) 2

(cotg[f(x)W = j 7{}, X \^ 2 = -fHx) ■ [1 + (cotg[f(x)]) 2 ]
(sin[f(x)\) 2

Aj.u£»JI Ajiillall Jlj^l

*JWIj ]-| , |[ JMJ- t#(x) UUIj [0 , tt] JMU* co5(x) =Uhitj[-| , |]
.l\$i« jSi 4au£«J! aJW! jlik-VI Oi*4 ^Vl o^j ^Mj j aauj JIjj ^ ]0 , n[ JW-JI Jc- cotg{x)

i4rcsin: 5H -> 5H

x -> i4rcsin(x) = a tq sin{a) = x
Arccos'.yi -» 91

x -> i4rccos(x) = b tq cos[b) = x
Arctg-.ft^ft .

x -> Arctgix) = c tq tg[c) = x
Arccotg:^,^^, .

x -» Arccotgix) = d tq cotgid) = x

Ijjli 4_ii*a£*JI 4JUI (jjliiwi (j->^ ' . '"'"*•

[Arcsin(x) ] x = , . = — — = . , = -7==: .

Urccos(x)] y = — r-— = — — — = , r f )17 = -7==

cosHb) -sin(b) Jl-[cos(b)] 2 Vl-x 2

[i4rct,g(x)] y = — -— = . , ... = - — -.

u tg\[c) l+[tg(c)] 2 1+x 2

[Arccotgix)]^ = ;

cotg\(d) -(l+[cotg(d)] 2 ) 1+x 2

Kcco S (/W)] ( - ^

[i4rccot,g(/(x))] =

\ _ ~[/(x)]\

l + [/U)] 2

(j^a*J JJ^-a £

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, f(x) = y <=> / _1 (y) = X Ai^J«JI ijjSj tSUli j 4_iui5L*JI Ajdllall JIjaII AiajJjj SjJ^jill jjjsll

. {x G Df tq — 1 < f{x) < 1} cr* i4rcsin[/(x)] i_sjj*j <c>a^

. (x G D^ tq — 1 < fix) < 1} csA Arccos[fix)] i_sjj*j 4c>a^

. D^ (^a i4rctg[/"(x)] i_jjj*j <cj^v>

. D^ tsA i4rccot,g[/(x)] <-±ij*i <c.>«^

AjjolLuil (_ya\j^.

QT ^Lajjj Ait ojji Jjii <Ub f (j£jl

. /Ma) = u^ a Ak. (jlii^U aIAS / <JIa!I cjj\£ li!

.]a,b[ J&- (jjlili!>U <L13 j [a , 6] ^Jb Sj^l^ 3Jb ^ jiU
.]a , b[ JMi<^ f ^s /\ AiiullAJUloli /(a) = /(&) o^ lij

]a , M ^ (jlili!>U Alii j [a , b] ^Je S j*!-- 4Jb / cjj\£ lil
.celaJl'VW -/(a) =/\(c)-[fc-a] u^

(>-i«Jj V g (jli* jj£ ^ ]a , Z?[ ^ (3lili!>U jjjLli j [a , b] t^ OPJ^** 00* f ,g cjj\£ lij
■ #M = f{ ,T f \ a \ Wx) - gla)] + f(a) -fix) + (3 UU\ J, Jjjj a,j^ J-i-3

x-o IgJLaxiail Ait j* ku3 Ift-ljlja Lai j A^i.j-aJI (JLaE-VI ^ji A V) a*^ I jj '— » ; jjjLciuij SjJJ^ dlVt rtn'unl UjJ <"<ljj!Viti sift

.■LjJj-aVl Jlj.l!l£ SjJuLva JJC- "lajjlaj Aijx-a JljJ

(Jlij^aJ oAtla 4JA^)J (j^-<y *J^^ AjjJaiJI 4Jajail_aJ

Aa.i jjSj iliia. ?-jjL. JL>- ^ jjUli^J j^IiS u^ f(x) , g(x) ^- h[x) = , , aJIa!I jSjJ

/(x)-^(a)

f \ { \

sHxl

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