[PDF] limite-et-continuité.pdf Montrer que f est constante.

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??? ??????? ???? ?????? x2[0;1]??k2N???? ???fk(x) =x?????x??? ?? ????? ??? ????f? ??? ??????? f????? ?? ????? ???? ???limx!0p1+xp1xx ???limx!+1xpx lnx+x???limx!0+xx lim x!1+lnx:ln(lnx)???limx!0(1 +x)1=x ???limx!11xarccosx ???limx!0x:sin 1x ???limx!+1xcosexx

2+1???limx!+1exsinx

???limx!+1x+arctanxx ???limx!0xb1=xc ???limx!+1xb1=xc f:x7! bxc+px bxc? ????f:R!R?????? ??? f(x) =(

1??x2Q

0??????

f(x) = sup n2Nx nn!? ??????? ???f????? ?? ?????? ????? ???? lim x!+1f(x)x =` <1? ??????? ????? ??????2[0;+1[??? ???f() =?

8x2I;f(x)=g(x)6= 0?

??????? ???f=g??f=g? ??????? ????? ??????c2[a;b]??? ??? p:f(a) +q:f(b) = (p+q):f(c)?

9(x1;y1)2I2;x1< y1??f(x1)f(y1)??9(x2;y2)2I2;x2< y2??f(x2)f(y2)?

'(t) =f((1t)x1+tx2)f((1t)y1+ty2) z2C7!zexp(z)2C? f ?? ??????? ???fn????? ?? ????? ???? ??????? ???f????? ????? ?? ????? ????

8x2[a;b];f(x)< g(x)?

??????? ????? ?????? >0??? ???

8x2[a;b];f(x)g(x)?

lim +1f= lim1f= +1? ??????? ???f????? ?? ??????? ??????? ????f:R!R?????? ??? f(x) =x1 +jxj? f(x)? ?????? ???? ??????? ?? ???? ????? ????? ? ??1=e?

8x2R;f0(x) =f(x+ 1)?

ff= Id?

8x2R;f(x) =f(x2)?

f x+ 12 =f(x)?

8x2R;f(2x) =f(x)cosx?

8x;y2R;f(x+y) =f(x) +f(y)?

??????? ??? ? ??????? r2Q?f(r) =ar????a=f(1)? ????f:R!R????? ??? ???? ????x;y2R? f(x+y) =f(x) +f(y)?

8x;y2R;fx+y2

=12 f(x) +f(y)?

8x2R;2f(x) =f(2x)?

?? ??????? ???f??? ??????

8x;y2R;fx+y2

=12 f(x) +f(y)? ?? ???? ???f(0) = 0? ???????

8x;y2R;f(x+y) =f(x) +f(y)?

u

0=a??8n2N;un+1=f(un)?

??????? ???un!0? '(t) = sup x2[0;1]f(x) +tg(x)? f k(x)fk1(x):::f(x)> x f(an)an??f(bn)bn? ?? ?????? ??? ??????(an)??(bn)?? ??????a0= 0??b0= 1? ??f(m)m?? ????an+1=m??bn+1=bn? ?????? ?? ????an+1=an??bn+1=m? f(an)f(c)f(bn) a nf(c)bn? f(c) =c? c= supx2[0;1];f(x)x? x=p10 k????p2Z??k2N f(x) = (1)p 1110

10??k2N?

????? x!0? p1 +xp1xx =1 +x(1x)x p1 +x+p1x=2p1 +x+p1x!1? ????? x!+1? xpx lnx+x=11=px lnxx + 1!1? ????? x!0+? x x= exlnx= eX ????X=xlnx!0????xx!1? ????? x!1+? lnx:ln(lnx) =XlnX ????X= lnx!0????lnx:ln(lnx)!0 ????? x!0? (1 +x)1=x= e1x ln(1+x)= eX ????X=ln(1+x)x !1????(1 +x)1=x!e? ????? x!1?

1xarccosx=1cosyy

=2sin2(y=2)y = sin(y=2)sin(y=2)y=2 ????y= arccosx!0????siny=2!0??siny=2y=2!1????1xarccosx!0? ????? x!0?xsin1x jxj !0? 2+ 1 xx

2+ 1!0?

????? x!+1? e xsinxex1!+1? ????? x!+1? x+ arctanxx

1arctanxx

2x!0? ????? x!0?

1=x1 b1=xc 1=x

b1=xc 1=x1 xb1=xc 1 jxj !0? ????? x!+1?1=x!0????b1=xc= 0????xb1=xc= 0!0? ????? x!0+?

Eb1=xc 1x

1!+1? ????? x!0+?

1=x1 b1=xc 1=x

1xxb1=xc 1

????xb1=xc !1? ?????x!0?

1=x1 b1=xc 1=x

1 b1=xc 1x

!0

0 b1=xc 1=x

?? ?????x!0?x2b1=xcx2 11x !0 1x

1 b1=xc 0?

?????? `?? ?????? ??g??+1?

8x2R;g(x) =g(x+nT)!n!+1`???? ??? ??????? ?? ?? ?????? ?g(x) =`?

?????? `?? ?????? ??f??+1? ??????x;y2]a;b[???? ???x < y? ????t2]x;y[? ?? ?f(t)f(y)? ?????t!x+? ?? ???????limx+ff(y)?? f(y)limy+f????limx+flimy+f? ?????x!a+?f(x) =bxc+px bxc !a=f(a)???E(x)!a? ?????x!a?f(x) =bxc+px bxc !a1 + 1 =a=f(a)???bxc !a1? ????a2R? ?? ?f(un) = 1!1??f(vn) = 0!0????f??? ??? ?? ?????? ??a?? ??? ???? u n(x) =xnn! ???????x??? ??? ???? u n+1(x)u n(x)=xn+ 1? ????n 0?????f(x)> x!x!+1+1?? ??? ??? ??????? ??????? lim +1f= infRf? ??8x2R;g(x)<0?????f(x)< x!x!11?? ??? ??? ??????? ??????? lim

1f= supRf?

g:x7!f(x)x ?? ????limx!+1g(x) =`? ???g() = 1????f() =? ????y?? ?????? ??? ?????? ??????? ?????f(a)??f(b)? ??????':I!R?????? ??? '(x) =f(x)=g(x)

8x2I;'(x)= 1?

y=pf(a) +qf(b)p+q? ??f(a)f(b)????? f(a) =pf(a) +qf(a)p+qypf(b) +qf(b)p+q=f(b)? ??????x0= (1t)x1+tx2??y0= (1t)y1+ty2? ????r >0??2R? ??????? zexp(z) =reiercos()+irsin()=rercos()ei(+rsin()) rercos()=R +rsin()[2]??? ??????? ?? ??????? ??????? ???sin()? ????2]0;[? ?? ???? r=+ 2ksin() f() =+ 2ksin()exp+ 2ksin()cos() + 2ksin()!+1???sin()!0+??+ 2k >0 exp+ 2ksin()cos() !+1???cos()!1? + 2ksin()!+1??exp+ 2ksin()cos() !0???cos()! 1? exp + 2ksin()cos() exp 12 + 2ksin()

0f()2Xexp(X)????X=12

+ 2ksin()!+1? ':R!R?????? ???'(x) =f(x)x???? ????x2R? ????fn1(c)? '(c) +'f(c)++'fn1(c) f(c)c+f2(c)f(c)++fn(c)fn1(c) ????T >0??? ??????? ??f? ???? ????x2R?xnT2[0;T]????n=E(x=T)????f(x)=f(xnT)M? ?????f??? ?????? ???M???R? ????M2R??? ???

8x2R;f(x)M?

???? ????x2R?f(g(x))M????fg??? ??????? ?? ???????M0? ???? ????x2R?g(f(x))M0???f(x)2[M;M]?????gf??? ??????? ??????': [a;b]!R?????? ??? '(x) =g(x)f(x) c2[a;b]? ??????='(c) =g(c)f(c)>0? ???? ????x2[a;b]?'(x)???? f(x)g(x)? ??????M=f(0) + 1? ???????lim+1f= lim1f= +1? ?? ??????A;B2R???? ???

8xA;f(x)M??8xB;f(x)M?

?? ?A0B???f(0)< M? ?? ?f(a)f(0)???02[A;B]????f(a)M? ???? ????x2[A;B]? ?? ?f(x)f(a)?? ???? ????x2]1;A][[B;+1[? f(x)Mf(a)? ?????f????? ?? ??????? ?????? ??a? ??? [0;+1[? f(x) =x1 +x= 111 +x ???]1;0[? f(x) =x1x=1 +11x y=f(x)()y=x1 +x()x=y1y? y=f(x)()y=x1x()x=y1 +y? ????x02]a;b[? ?? ?limaf < f(x0)0? ????y+2]f(x0);f(x0) +"]\]limaf;limbf[? ?? ??????x+2]a;b[ ??? ???f(x+) =y+? ????y2[f(x0)";f(x0)[\]limaf;limbf[? ?? ??????x2]a;b[??? ??? f(x) =y? ???? ????x2]a;b[? ??jxx0j ?????xxx+????yf(x)y+????f(x)f(x0)"? ????? ????f?? ?? ????? f(x) = ex? ????2[0;1=e]? ?? ??????2R+??? ???e=?? ???? ?? ?????? ??? f(f(x))< f(x) ?? ????f= Id?

8x2R;f(x) =f((x)2) =f(x2) =f(x)

????f??? ?????? f(x1=2n) =f(x1=2n1) ==f(x)quotesdbs_dbs47.pdfusesText_47

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