Travaux dirigés-feuille 1 : Révisions

Travaux dirigés-feuille 1 : Révisions
Exercices
x7→ f(x)
f(x) = ln(x+ 1) f(x) = ln(|x|+ 1) f(x) = ln(|x+ 1|)
f(x) = √x2+x f (x) = pcos(x)f(x) = tan(x2+ 1)
f(x) = √2x+1−3
√5−x−1f(x) = (1 + x)x
1. lim
x→+∞
x4+x−1
2x4−x2. lim
x→1
x2+x−2
x2+ 2x−33. lim
x→0
√x+ 1 −1
x
4. lim
x→0
sin(x)
x5. lim
x→0
ln(x)
√x6. lim
x→0
ex
1 + x5
7. lim
x→0
ex2−1
sin2(x)8. lim
x→01
ln(x)−1 x−19. lim
x→0
1−cos(x)
x2
10. lim
x→0(1 + 1
x2)x
1. f1(x) = ln(x4) 2. f2(x) = ln4(x) 3. f3(x) = ln(4x)
4. f4(x)=(x2−2x)ex5. f5(x) = √x2−2 6. f6(x) = si n(x) + cos(x)
si n(x)−cos(x)
7. f7(x) = e
x3
28. f8(x) = √ex39. f9(x) = e√3+cos x
10. f10(x) = x2+ 1
(x+ 1)3
0
(|xn|)n∈Nl(xn)n∈Nl
−l
(xn)n∈Nl(x2n+n2)n∈Nl
(xn−yn)n∈N0 (xn)n∈N(yn)n∈N
0
0
0
1
un=n2+3n+1
1+n2+n3∀n∈N
un=|sin(n)|
n+1 ∀n∈N
un=2n
n5+3n+1 ∀n∈N
un= (n4+ 1)(n3+ 2)e−n∀n∈N
un=log(n)
n+1 ∀n∈N
un=√n2+n+ 1 −√n2−n+ 1 ∀n∈N
un=pn+√n2+ 1 −pn+√n2−1∀n∈N

Travaux dirigés-feuille 1 : Révisions
un=n−√n2+1
n−√n2−1∀n∈N
(un)n∈N
un+1 =un+b∀n∈N,
u0=x,
b x ≥0un=bn +x∀n∈N
(un)n∈N
un+1 =aun∀n∈N,
u0=x,
a x ≥0
un=xan∀n∈N
a=1
2x= 1000
un
1
a= 2 x= 2
un
(un)n∈N
un+1 =aun+b∀n∈N,
u0=x,
a6= 1 b x ≥0
un=an(x+b
a−1)−b a−1∀n∈N
a= 2 x= 10 b
(un)n∈N
un+1 =f(un)un∀n∈N,
u0=x,
x≥0f: [0, ∞)→[0, 1]
f(y) = (a y <2,
2a
yy≥2.
(un)n∈N
x= 3 a=1
20<un<2∀n≥1
x= 1 a= 2
(un)n∈N
un+1 =f(un)un∀n∈N,
u0=x,
x≥0f: [0, ∞)→[0, 1]
f(y) = (a y <2,
0y≥2.

Travaux dirigés-feuille 1 : Révisions
(un)n∈N
a=1
2x≥0a= 1 x= 1
a= 1 x= 3 a= 2 x≥0
x= 1
a=9
10
a=11
10
(un)n∈N
un+1 =aun
(1+un)b∀n∈N,
u0=x,
a,b x ≥0
a=1
2b x
un≥0∀n∈N(un)n∈N
lim
n→∞un= 0
a= 2 b= 1 x= 2 lim
n→∞un= 1
a= 2 b= 1 x=1
2lim
n→∞un= 1
a= 2 b=1
2x= 2 lim
n→∞un= 3
(un)n∈N
un+1 =2un
1+un
un
un+s∀n∈N,
u0= 1,
s
s≥2un∈[0, 1] ∀n∈N
s≥2 (un)n∈N
s≥2
n un
un+2 =un+1 +un∀n∈N,
u0= 0,
u1= 1.
un=1
√5((1+√5
2)n−(1−√5 2)n)∀n∈N
un+2 =un+1 +un∀n∈N
(un)n∈N
un+1 = 2un(1 −bun)∀n∈N,
u0=x,
b>0x≥0
x∈ {0, 1
b}lim
n→∞un= 0
x∈]0, 1
2b]lim
n→∞un=1
2b
x∈[1
2b,1 b[lim
n→∞un=1
2b
(un)n∈N
un+1 =un−unln(un)∀n∈N,
u0=x,
x∈]0, e[lim
n→∞un= 1
(un)n∈N
(un+1 =au2
n
u2
n+b2−E un∀n∈N,
u0=x,
a>0b E >0

Travaux dirigés-feuille 1 : Révisions
E= 0 a2>4b2
t7→ at2
t2+b20<x<a−√a2−4b2 2
0<aun
u2
n+b2<1∀n∈N.
E= 0 a2>4b20<x<a−√a2−4b2
2(un)n∈N
0
a>2b= 1 E>(a−2)/2
x>0 0