Applications
Applications des Limites
Exercice 1
Calculer lim(x→0) sin(3x)/(2x)
Solution
sin(3x)/(2x) = (3/2) × sin(3x)/(3x)
Quand x→0, 3x→0, donc sin(3x)/(3x) → 1
Limite = 3/2
Exercice 2
Calculer lim(x→0) (1 - cos(x))/x²
Solution
On utilise : 1 - cos(x) = 2sin²(x/2)
(1 - cos(x))/x² = 2sin²(x/2)/x² = (1/2) × (sin(x/2)/(x/2))²
Quand x→0, sin(x/2)/(x/2) → 1
Limite = 1/2
Exercice 3
Calculer lim(x→0) tan(x)/x
Solution
tan(x)/x = sin(x)/(x×cos(x)) = (sin(x)/x) × (1/cos(x))
Quand x→0 : sin(x)/x → 1 et 1/cos(x) → 1
Limite = 1
