Chapitre 5

+∞
f[A, +∞[
`∈Rf ` +∞lim
x→+∞f(x) = `
I ` B x > B f(x)∈I
f(x)`
+∞
f+∞+∞lim
x→+∞f(x) = +∞C
B x > B f(x)> C f(x)
+∞
f−∞ +∞lim
x→+∞f(x) = −∞ C
B x > B f(x)< C f(x)
+∞
α∈R
lim
x→+∞xα=
0α < 0,
1α= 0,
+∞α > 0.
lim
x→+∞ex= +∞
lim
x→+∞ln(x) = +∞
sin +∞
0 1 +∞ −∞
` I ` 0
1n∈Zsin(2nπ) = 0 sin(2nπ +π/2) = 1
B x > B sin(x)∈Isin `
−∞
f]− ∞, A]
`∈Rf ` −∞ lim
x→−∞ f(x) = `
I ` B x < B f(x)∈I
f(x)`
−∞
f+∞ −∞ lim
x→−∞ f(x) = +∞C
B x < B f(x)> C f(x)
−∞
f−∞ −∞ lim
x→−∞ f(x) = −∞ C
B x < B f(x)< C f(x)
−∞

n
lim
x→−∞ xn=(+∞n ,
−∞ n .
lim
x→−∞
1
xn= 0
lim
x→−∞ ex= 0
f D a
D f a D
f a
`∈Rf ` a lim
x→af(x) = `
I ` J a
x∈J∩D f(x)∈I f(x)
` a
f+∞alim
x→af(x) = +∞C
J a x ∈J∩D f (x)> C
f(x)`
a
f−∞ alim
x→af(x) = −∞ C
J a x ∈J∩D f (x)< C
f(x)`
a
lim
x→0
sin(x)
x= 0
lim
x→0
1
x2= +∞
a
]a, b[a < b f
`∈Rf ` a lim
x→a+f(x) =
` I ` J a
x∈J∩D x > a f(x)∈I f(x)
` a a
f+∞alim
x→a+f(x)=+∞
C J a x ∈J∩D x > a f(x)> C
f(x)`
a a
f−∞ alim
x→a+f(x) = −∞
C J a x ∈J∩D x > a f(x)< C
f(x)`
a a
lim
x→0+
1
x= +∞lim
x→0−
1
x=−∞

+∞ −∞ a a+a−f g
D
lim f(x) lim g(x) lim f(x) + g(x)
`∈R`0∈R`+`0
`∈R+∞+∞
`∈R−∞ −∞
+∞+∞+∞
−∞ −∞ −∞
+∞ −∞ F I
lim f(x) lim g(x) lim f(x)g(x)
`∈R`0∈R``0
` > 0 +∞+∞
` > 0−∞ −∞
` < 0 +∞ −∞
` < 0−∞ +∞
0±∞ F I
+∞+∞+∞
+∞ −∞ −∞
−∞ +∞ −∞
−∞ −∞ +∞
g D
lim f(x) lim g(x) lim f(x)/g(x)
`∈R`06= 0 `/`0
` > 0 0++∞0
` > 0 0−−∞0
` < 0 0+−∞0
` < 0 0−+∞0
`∈R±∞ 0
+∞` > 0 0++∞
+∞` < 0 0−−∞
+∞` > 0 0+−∞
+∞` < 0 0−+∞
0 0 F I
±∞ ±∞ F I
f I g J f(I)⊆J
a I lim
x→af(x) = ` `
J
lim
x→ag◦f(x) = lim
y→`g(y).
a ` +∞ −∞
lim
x→+∞ln(2 + e−x)y=−x
lim
x→+∞2 + e−x= lim y→−∞ 2 + ey= 2.
z= 2 + e−x
lim
x→+∞ln(2 + e−x) = lim
z→2ln(z) = ln(2).

f g ]A, +∞[
x > A f(x)≤g(x)
lim
x→+∞f(x) = `lim n→+∞g(x) = `0`≤`0
lim
x→+∞f(x) = +∞lim x→+∞g(x)=+∞
lim
x→+∞g(x) = −∞ lim x→+∞f(x) = −∞
`0< ` a < `
`0< b < ` x g(x)∈]a, b[g(x)< b f(x)< b x
]b, ` + 1[ f(x)x
f+∞`≤`0
A∈Rx f(x)> A g(x)≥f(x)> A g +∞
+∞
2
−∞ a a+a−
x∈R
x−1≤x+ cos(x)≤x+ 1.
lim
x→+∞x+ cos(x)=+∞lim x→−∞ x+ cos(x) = −∞
f g h
]A, +∞[x > A f(x)≤g(x)≤h(x)
lim
x→+∞f(x) = lim x→+∞h(x) = `lim x→+∞g(x) = `
I ` x f(x)
h(x)I I f(x)≤g(x)≤h(x)g(x)∈I
g ` 2
−∞ a a+a−
+∞x+cos(x)
xx+cos(x)≥x−1
x x + cos(x) +∞ ∞/∞
x > 0
1−1
x=x−1 x≤x+ cos(x) x≤x+ 1 x= 1 + 1 x.
x+cos(x)
x1 +∞
α > 0n > 0
lim
x→+∞
ex
xα= +∞,lim
x→+∞
ln(x)
xα= 0+,lim
x→0+xαln(x)=0−,lim
x→−∞ xnex= 0.
fRf(x) = ex−x
x∈Rf0(x) = ex−1f0(x)≥0x≥0f0(x)≤0x≤0f

f0x∈Rf(x)≥f(0) = 1 ex≥x+ 1
ex/2≥x
2+ 1 ≥x 2x≥0ex≥x2 4x≥0
ex
x≥x 4.
ex
x+∞
lim
x→+∞
ex/α
x/α = lim
x→+∞αex/α
x= +∞.
1/α
lim
x→+∞
ex/α
x= +∞.
y=ex/α
x
lim
x→+∞
ex
xα= lim
x→+∞ ex/α
x!α
= lim
y→+∞yα= +∞.
y=αln(x)
lim
x→+∞
ln(x)
xα= lim
x→+∞
1
α
αln(x)
eαln(x)= lim
y→+∞
1
α
y
ey.
`/ +∞`= 1/α 0+
y= 1/x x 0+y+∞
lim
x→0+xαln(x) = lim x→0+−1
x−α
ln(1/x) = lim
y→+∞−y−αln(y) lim y→+∞−ln(y)
yα= 0−.
y=−x x −∞ y+∞
lim
x→−∞ xnex= lim x→−∞(−1)n(−x)n1
e−x= lim
y→+∞(−1)nyn
ey.
`/ +∞`= (−1)n
n2
+∞ex
x6+ ln(x)2
ex
x6+ ln(x)2=ex x6
1
1 + ln(x)
x32.
lim
x→+∞
ex
x6= +∞,lim
x→+∞
ln(x)
x3= 0.
lim
x→+∞ln(x)
x32
= lim
y→0y2= 0.
lim
x→+∞
ex
x6+ ln(x)2= +∞.