# Student Suppl. for Calculus of One and Several Vars by S. Salas, E. Hille, By S. Salas, E. Hille,

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Example text

10 tend vers z4ro Remarquons que la condition (iii) entraZne que quand t tend vers l'infini. nti~re d'ordrg pres%ue iDf@rieur il existe alors une oonstante z E Cn Q K > O ~ telle que Sup ' o~ f Log Icl ~ ~ Q(211dl) If(z + ~)1 ~ - ~ Q(211~II) est la fonction intervenant dans la d6finition plus haute~ et off le Sup peut 6tre pris dans D6monstration. ~n si z 6 ~n . - I1 existe une constante ~our tout t ~ f(O) = I . > fz(t) = f(tz) . A > 0 t. 3 ordre qui v6rifie telle que n(=, ~) -< A M~(2 ll~ll) r > 0 D4monstration.

IlR~ =11) quantit~ est born@e. (=) -M En effet~ @tant donngs~ la suite M(p) ~. 3 e% , [IzIl(~j(1 + llzfl)) ~ "3 E x p ( - M ( ~ ) - 6Z E x

E. ~ est cons4quence de (a) ~o(~p~,~ J ~) = ~ EU JI 8(N(p), n) Montrons oe fait• f tout compact K C Q , ~i et e% tout telle que~ quel que soit Ipl = ~ 6 ~ = A I alors la condition (c)(ii) que l'on a h > 0 8o(M(p), O) , on sait donc ~ e , , il correspond une constante AK~ h (p) 6 ~n K c O K~ IM de compacts contenus dans est contenu dans l'un des , on peu% supposer que la s ~ t e K~ ~ ...... > ~ 5, Z croissante. ) Consid4rons alors une suite croissante ~ . D'une ~[(p) est dgfinie par une I Posons (c)(i) .