Analyse asymptotique

∀(un)∈RNun=o(1
n) =⇒un→0
∀(un)∈RNun=o(ln(n)) =⇒un→0
∀(un)∈RNun+1 ∼un
∀(un)∈RNun+1 ∼un=⇒eun+1 =O(eun)
2n
n∼22n
√nπ
f:]0,+∞[→Rf(x) = x−ln(x)
n∈N∗(un, vn)∈]0,1] ×
[1,+∞[f(un) = f(vn) = n
(un)n∈N∗0un∼e−n
(vn)n∈N∗+∞vn∼n
nsin 1
n3
e1/n −1,√1 + nln 1 + 1
n2
r1 + 1
n−1!+ sin 1 n2,1
p(n+ 1)3+n1/n −1.
0
0
ln(1 + x)−x
ex−1;
sin(x)
x−1
x
o(x) + o(x2)
√x+o(x);o(x2)+3x3 √1 + x2−1.
(1 + n)−n
0 3
a:x7→ 2 ln(1 −x)b:x7→ 2 ln(1 −x2)c:x7→ sin(x)
x
d:x7→ 1−cos(x)
x2e:x7→ 1
2−xf:x7→ (1 + x)1
x
4x7→ exp 1
cos(x).
x7→ ln
99
X
k=0
xk
k!
cos
5π
6
tan(x)
3
x7→ arctan(x) arctan(x) =
x→0a0+a1x+a2x2+a3x3+o(x3)
a0a2a1
a3tan ◦arctan = R
lim
x→0+
ln(1 + x)−x
x2et lim
n→+∞1 + 1
nn

lim
x→0
ex2−cos(x)
x2; lim
x→−∞ px2+ 5x+1+x.
a b R∗
+a b
lim
x→+∞ax+bx
21
x
lim
x→0+ ax+bx
21
x
.
2
g:x7→ ln sin(x)
x
lim
x→0
g(x)
x2lim
n→+∞nsin( 1
n)n2
x7→ 2ex−√1+4x−√1+6x20
x7→ cos(x)sin(x)−cos(x)tan(x)0
x7→ √x2+ 1 −23
√x3+x+4 √x4+x2+∞
DL 3 0
f:
]0, π]→R
x7→ ln 3sin(x)
x+e−x.
f0
f(0) f0(0)
0
f:−π
2,π 2→R
f(x) = (2
sin2(x)+1
ln(cos(x)) x∈0,π
2
1x∈−π
2,0
f0
f
0
f:x7→ arctan x
(sin(x))3−1 x2
g
0
0