Fonctions d`une variable réelle.

TSI1
n∈N∗
a) lim
x→0
x>0
√x= 0 b) lim
x→+∞
xn= +∞c) lim
x→0
x>0
1
xn= +∞
f∈ F(I, R)f `
a I f a
f∈ F(I, R)f ` > 0
a I f > 0a
f, g ∈ F(I, R)a
f=o(g)⇒f=O(g)
f1=o(g1)
f2=O(g2)⇒f1f2=o(g1g2)
f=o(g)
g=O(h)⇒f=o(h)o(sin x)o(x)x→0
f g f(x)∼
x→ag(x)
ln f∼
aln g
f g a 0 +∞ln(f)∼
aln(g)
exp f∼
aexp g f g a ±∞
f:x→π
2−arcsin x x+ 1
f+∞f(x)
x→+∞
a) lim
x→0
1−cos x
x(2 −x) tan(2x)b) lim
x→0
(1 −ex) sin x
x2+x3c) lim
x→1
2
(2x2−3x+ 1) tan(πx)
d) lim
x→π
2
tan xtan(2x) e) lim
x→0
ln(cos(ax))
ln(cos(bx)) a, b b 6= 0

a) lim
x→0
ln(1 + 3x)
xb) lim
x→0
ex2−1
xc) lim
x→0
e5x−e3x
xd) lim
x→0
x
esin x−1
e) lim
x→0
5x−2x
xf) lim
x→+∞1 + 1
xx
g) lim
x→0
sin x
tan xh) lim
x→0
x
sin x+x2
i) lim
x→0
xsin x
1−cos xj) lim
x→+∞
cos x
xk) lim
x→0
(1 −ex)(1 −cos x)
x3+x4l) lim
x→−1
x5+ 1
x3+ 1
m) lim
x→a
xn−an
xp−ap(a∈Rp, n ∈N∗) n) lim
x→1
√x+ 3 −2
x−1o) lim
x→0
x
√x2+ 1 −1
p) lim
x→a
x−a
√x−√aq) lim
x→+∞px2+x−x+ 1
n∈Nlim
t→π
sin((n+ 1)t)
sin t
f:x7→ p|x−1|e1
xx6= 0 f(0) = 0
f:x7→ x2
1−exx6= 0 f(0) = 0
f:x7→ x
|x|x6= 0 f(0) = 0
f:x7→ 1x∈Q
0x∈R\Q
f(x) = xsin 1
xx6= 0 f(0) = 0 fR
x7→ √xR+
0a6= 0
a) f:x7→ xsin x
x2−1b) g:x7→ xx+1 c) h:x7→ √1 + x−1 x
fR R x y
(∗)f(x+y) = f(x) + f(y)
(R,+) f(1) = a
f(∗)f(0) = 0 f
n x f(nx) = nf (x)f(n) = na
r f(r) = ar
f(1
q) = a qq∈N∗

TSI1
x f(x) = ax
Q R
fR
lim
x→−∞
f(x) = −∞ lim
x→+∞
f(x) = +∞
f(x) = 0
R[X]
f[a, b]Ra<b
f x0∈[a, b]f(x0) = x0
x0
g g(x) = f(x)−x
fR0
∀x∈R, f(x) = f(2x)
fR
f I f(I)
f f
f[a, b]x[a, b]
f(x)>0m > 0x[a, b]f(x)>m
f g [a, b]x[a, b]
f(x)> g(x)m > 0x[a, b]
f(x)>g(x) + m
fR R f(0) = 0, f(1) = 1
x f ◦f(x) = x
f f
x[0,1] f(x) = x
IRf g I
∀x∈I, f2(x) = g2(x)6= 0
f=g f =−g
fR+∞ −∞
f