Techniques d`approximation

x a
x
√1 + x√1 + x'1 + x/2
√1 + x'1 + x/2−x2/8
'
x
h(x)
f(x)f(x)
f(x) + h(x)h(x)
f(x)a±∞ lim
x→ah(x)/f(x) = 0
lim
x→∞ h(x)/f(x) = 0 h¿f
h¿
af h(x)¿f(x)¿h f
x¿
+∞x2x¿ +∞−x2
f¿glim f= 0
f/g
1
x¿
+∞1¿ +∞ln x¿ +∞x¿ +∞x2¿ +∞... ¿ +∞ex
xn¿
0. . . ¿ 0x2¿ 0x¿ 01¿ 0ln x¿ 0
1
x;
f¿g g ¿h⇒f¿h
lim f/h = lim f/g ×g/h
h f f f +h
f∼g
f∼
ag f g f g
alim
af/g = 1

g= 0 f∼
ag
ε(x)f(x) = g(x)(1 + ε(x)) lim
x→aε(x) = 0
ε(x)
∼f∼f f ∼g⇐⇒ g∼f f ∼g
g∼h⇒f∼h¿h¿f
f∼g⇒h¿g
x2¿
∞x3x2∼
1x3x3¿
0x2
f1∼f2;g1∼g2⇒f1g1∼f2g2;f1/g1∼f2/g2
f¿g⇐⇒ f+g∼g f1¿g1;f2¿g2⇒f1+g1
f2+g2∼g1 g2
f1∼f2⇒fα
1∼fα 2α∈R
f1∼f2⇒f1+g∼f2+g x −x2∼
∞1−x2
(x−x2)+x2/∼
∞(1−x2)+x2
f
f=f0+h h h ¿f
h
h=o(f)
g=o(f)g¿f x
o(f)
fsin x=x+o(x2)
sin x−x x2lim
x→0
sin x−x
x2= 0
o(f)
o(f) +
o(f) = o(f)
f f
o(f)f.εif(x)εi(x)
lim εi= 0 i . . .

f g +∞
x f g
g f +∞
f g f =O(g)∃A, x0,∀x > x0,|f(x)|< A|g(x)|
A > 0
o(g)
sin x=x+o(x2)
sin x=x+O(x3) sin x=
x−x3/6 + o(x3)
O(nk)n k
ana
n
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¿+∞xn1¿
∞x¿ ∞x2···
xn¿···¿x2¿x¿1¿1
x¿1 x2¿···
x+∞
ln x¿xn¿2x¿ex¿xx
xα(ln x)β
xα

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f cf0
c6= 0 f0f∼cf0
cf0∼c0f1f0¿f1cf0∼df0⇒c=d
xn+∞P(x) =
n
P
k=0
akxk
anxnsin x ex. . .
lim f/fif fi
fk
c f cfk
f cf0f=cf0+o(f0)
cf0df0
. . .
f∼cf0g∼df1lim f/g = lim(c/d)(f0/f1)
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f
f−cf0
c0f1f1¿f0f=cf0+c0f1+o(f1)
f=cf0+o(f0)
f=cofo+c1f1+···cnfn+o(fn)
fifn¿fn−1¿···f1¿f0) ;
f
xn
f n
n(f)nP(x)≤n
f(x)−P(x) = o(xn)P(x) =
n
P
k=0
akxkf(x) = a0+a1x+···+anxn+o(xn)
n(f) = P
f(x) = P(x) + o(xn)
P n P
m n
min(m;n)
n

Cn
n
o(xn)
xnx
f(x) = 1+x+1015xp|x|1
f(x) = 1+x+o(x)x/10
|x|<10−32
1/(1 ±x)
1
1−x= 1 + x+x2+··· +xn+o(xn)
1
1 + x= 1 −x+x2+··· + (−1)nxn+o(xn)
ln(1 + x) = x−x2
2+x3 3− ··· +(−1)n+1xn n+o(xn)
ex= 1 + x+x2
2+x3 6+··· +xn n!+o(xn)
sin x=x−x3
6+x5 120 − ··· +(−1)nx2n+1 (2n+ 1)! +o(x2n+2)
cos x= 1 −x2
2+x4 24 − ··· +(−1)nx2n (2n)! +o(x2n+1)
(1 + x)α= 1 + αx +α(α−1)
2x2+···+α(α−1)(α−2) ···(α−n+ 1) n!xn+o(xn)
sh x=x+x3
6+x5 120 +··· +x2n+1 (2n+ 1)! +o(x2n+2)