Bonjour,
1) Df = R - {-2,2}
lim f(x) en + ou - infini = lim x³/-x² = lim x = +/-∞
lim f(x) en -2+/- = lim -8/(4 - x²) = -/+ ∞
lim f(x) en 2+/- = lim 8/(4 - x²) = -/+ ∞
2) Asymptotes verticales : x = -2 et x = 2
lim en +/- ∞ f(x)/x = lim x³/-x³ = -1
donc branche parabolique de pente -1
3) f'(x) = 1/(4 - x²)² * [(3x² - 6x)(4 - x²) - (x³ - 3x² + 12)(-2x)]
= 1/(4 - x²)² * [12x² - 3x⁴ - 24x + 6x³ + 2x⁴ - 6x³ + 24x]
= (-x⁴ + 12x²)/(4 - x²)²
= x²(12 - x²)/(4 - x²)²
x -∞ -2√3 -2 2 2√3 +∞
f'(x) - 0 + || + || + 0 -
f(x) ...
