Théorèmes de comparaison

FondamentalThéorème

α %alpha désigne un réel fini, + ∞ + infinity ou − ∞ - infinity ,

  1. Si pour x x voisin de α %alpha , f ( x ) ≤ g ( x ) f(x)<=g(x) et lim x → α f ( x ) = + ∞ lim from {x toward %alpha} f(x) = +infinity alors : lim x → α g ( x ) = + ∞ lim from {x toward %alpha} g(x) = +infinity

  2. Si pour x x voisin de α %alpha , f ( x ) ≤ g ( x ) f(x)<=g(x) et lim x → α g ( x ) = − ∞ lim from {x toward %alpha} g(x) = -infinity alors : lim x → α f ( x ) = − ∞ lim from {x toward %alpha} f(x) = -infinity

  3. Si pour x x voisin de α %alpha , u ( x ) ≤ f ( x ) ≤ v ( x ) u(x)<=f(x)<=v(x) et lim x → α u ( x ) = lim x → α v ( x ) = l lim from {x toward %alpha} u(x) = lim from {x toward %alpha} v(x) = l alors : lim x → α f ( x ) = l lim from {x toward %alpha} f(x) = l

Exemple

Déterminer lim x → + ∞ cos x x 2 lim from {x toward +infinity} {{cos x}over{x^2}}

On a − 1 x 2 ≤ cos x x 2 ≤ 1 x 2 -{{1}over{x^2}}<={cos x}over{x^2}<={1}over{x^2} donc lim x → + ∞ cos x x 2 = 0 lim from {x toward +infinity} {{cos x}over{x^2}}=0