Limites de suites, cours, terminale S - MathsFG

(un)
(un)ll
A l (un)
N n ≥N un∈A
l
limn→+∞un=l
(un)l
(un)l l0l
l0< l l0> l e
l0+e<l−e(un)l N
]l−e;l+e[
]l0−e;l0+e[l0
N
l0
(un) (un)p un+1 =f(un)n≥p
l n
e l
p upl e
u up
n p
|u−l|< e
u f(u)
n n + 1
n
(un)un+1 =u2
nn u1= 0,75 n p
(un)
l= 0 e er e

P U E L
abs(U−L)> E
P+ 1 BP
U2BU
P
P→P
U→U
L→L
E→E
Abs(U−L)> E
P+ 1 →P
U2→U
P P
(1
n)1
√n)1
npp
•limn→+∞1
n= 0
•limn→+∞1
√n= 0
•limn→+∞1
np= 0
•a
1
N< a (1 n)
n > N 1
n< a 1 n
•a
N1
√N≤a√N≥1
aN≥1 a2N
N≥1
a2
1
√N≤a(1 √n)
n≥N1
√n≤a
•
•
•(un) +∞+∞
−∞ [a; +∞[a
]− ∞;a]
limn→+∞un= +∞
limn→+∞un=−∞

(sin(n))
•limn→+∞n= +∞
•limn→+∞√n= +∞
•limn→+∞np= +∞p
•m p limn→+∞mx +p=signe(m)∞
•]a; +∞[N N > a n
n≥N n ]a; +∞[ +∞
•]a; +∞[N√N > a N > a2
N N > a2√N > a n ≤N√n > a
(√n) +∞
•
(un)
•k(un)l
(kun)kl
•k(un) +∞ −∞ kun
signe(k)∞ −signe(k)∞
•A kl a > 0 ]kl −a;kl +a[
A(un)l N
n≥N un∈]l−a
k;l+a k[kl n ≥N
kun∈]kl −a;kl +a[kl
•a > 0 (un) +∞N
n≥N un>a
kn≥N kun> a
lim unl+∞+∞ −∞ l l
lim vnl0+∞ −∞ −∞ +∞ −∞
lim un+vnl+l0+∞+∞ −∞
lim un×vnll0+∞ −∞ +∞l∞l6= 0 l∞l6= 0
lim un
vnl06= 0 l l0

•unun=3n4+n
n3n≥1un= 3n+1 nlimn→+∞= +∞
limn→+∞1
n= 0 (un) +∞
•vnvn= 5n2−6n n ≥0vn=n(5n−6) limn→+∞n= +∞
limn→+∞5n−6 = +∞(vn) +∞
•(un) +∞(vn)
vn≥un(vn)
+∞
•(un)−∞ (vn)
vn≤un(vn)−∞
]a; +∞[a(un) +∞
N(un) ]a; +∞[
N0vn≥unN N0(vn) ]a; +∞[
(vn) +∞
u v w v w
l vn≤un≤wnu
l
l e
]l−e;l+e[ (vn)l
N vnl−e(wn)N0
wn< l +e N00 vn≤un≤wn
N N0N00 un≤vn> l −e un≥wn≤l+e wn
l
(un)l
(un)l