# A first course in calculus, Fifth Edition by Serge Lang By Serge Lang

This 5th variation of Lang's ebook covers all of the themes generally taught within the first-year calculus series. Divided into 5 components, each one component of a primary path IN CALCULUS comprises examples and functions when it comes to the subject coated. furthermore, the rear of the publication includes distinct suggestions to plenty of the routines, permitting them to be used as worked-out examples -- one of many major advancements over earlier variations.

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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) .