Calculs de DL

Calculs de DL

Fonctions trigonométriques f(x) a n DL

x/sinx 0 4 1 +x²/6 + 7x⁴/360 1/cosx 0 4 1 +x²/2 + 5x⁴/24

ln(sinx/x) 0 6 −x²/6−x⁴/180−x⁶/2835 exp(sinx/x) 0 4 e(1−x²/6 +x⁴/45)

√

tanx π/4 2 1 +h+h²/2

sin(x+x²+x³−x⁴) 0 4 x+x²+ 5x³/6−3x⁴/2 ln(xtan(1/x)) ∞ 4 x⁻²/3 + 7x⁻⁴/90 (1−cosx)/(e^x−1)² 0 2 1/2−x/2 +x²/6

sin((πcosx)/2) 0 6 1−π²x⁴/32 +π²x⁶/192 cosxln(1 +x) 0 4 x−x²/2−x³/6

(sinx−1)/(cosx+ 1) 0 2 −1/2 +x/2−x²/8

ln(2 cosx+ tanx) 0 4 ln 2 +x/2−5x²/8 + 11x³/24−59x⁴/192 e^cosx 0 5 e(1−x²/2 +x⁴/6)

Fonctions circulaires inverses f(x) a n DL

arcsin²x 0 6 x²+x⁴/3 + 8x⁶/45 1/arcsin²x 0 2 x⁻²−1/3−x²/15 arctanp

(x+ 1)/(x+ 2) ∞ 2 π/4−x⁻¹/4 + 3x⁻²/8 arccos(sinx/x) 0 3 |x|/√

3(1−x²/90)

1/arctanx 0 5 x⁻¹+x/3−4x³/45 + 44x⁵/945 arcsin√

x 1/4 3 π/6 + 1/√

3(2h−4h²/3 + 32h³/9) arcsin(sin²x) 0 8 x²−x⁴/3 + 19x⁶/90−107x⁸/630 arctan(1 +x) 0 4 π/4 +x/2−x²/4 +x³/12 arcsinx/(x−x²) 0 2 1 +x+ 7x²/6

e^arcsinx 1/2 2 e^π/6(1 + 2h/√

3 + 2(1 +√

3)h²/(3√ 3)) e^1/xarctanx ∞ 3 ^π₂ + (^π₂ −1)x⁻¹+ (^π₄ −1)x⁻²+ (₁₂^π −¹₆)x⁻³

dl.tex – mardi 5 juin 2018

Exponentielle et logarithme f(x) a n DL

x/(e^x−1) 0 2 1−x/2 +x²/12 lnx/√

x 1 3 h−h²+ 23h³/24 ln((2−x)/(3−x²)) 0 2 ln(2/3)−x/2 + 5x²/24 ln(1 +x)/(1−x+x²) 0 3 x+x²/2−x³/6

chx/ln(1 +x) 0 1 x⁻¹+ 1/2 + 5x/12 ln(ln(1 +x)/x) 0 3 −x/2 + 5x²/24−x³/8

ln(a^x+b^x) 0 2 ln 2 +xln√

ab+x²ln²(a/b)/8 exp(1/x)/x² 1 3 e(1−3h+ 13h²/2−73h³/6) Fonctions hyperboliques inverses

f(x) a n DL

argth(sinx) 0 5 x+x³/6 +x⁵/24 argsh(e^x) 0 2 ln(1 +√

2) + 1/√

2(x+x²/4) Formes exponentielles

f(x) a n DL

(1−x+x²)^1/x 0 2 e⁻¹(1 +x/2 + 19x²/24)

((1 +x)/(1−x))^α 0 3 1 + 2αx+ 2α²x²+ 2α(2α²+ 1)x³/3 (sinx/x)^2/x2 0 3 e^−1/3(1−x²/90)

(sinx/x)^3/x2 0 4 e^−1/2(1−x²/60−139x⁴/151200) (1 + sinx)^1/x 0 2 e(1−x/2 + 7x²/24)

(1 + sinx+ cosx)^x 0 2 1 +xln 2 +x²(ln²2 + 1)/2 (sinx)^sinx π/2 4 1−h²/2 + 7h⁴/24

(tanx)^{tan 2x} π/4 4 e⁻¹(1 + 2h²/3 + 4h⁴/5) Développer d’abord ln((1 +x)/(1−x)) Radicaux

f(x) a n DL xp

(x−1)/(x+ 1) 2 3 1/√

3(2 + 5h/3 +h³/54) p1 +√

1−x 0 3 √

2(1−x/8−5x²/128−21x³/1024) p1−√

1−x² 0 5 |x|/√

2(1 +x²/8 + 7x⁴/128) e^x−√

1 + 2x 0 5 x²−x³/3 + 2x⁴/3−13x⁵/15 (√³

x³+x²+√³

x³−x²)/x ∞ 3 2−2x⁻²/9

Exercice 1. EIT 1999

Calculer le développement limité de

tanx x

1/x²

en 0 à l’ordre 3.

dl.tex – page 2

solutions

Exercice 1.

e^1/3

1 + 7

90x²+≤(x³)

.

dl.tex – page 3