Limite Fundamentale

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Limite fundamentale lim

(1+ f(x)) 1/f(x)= e

dacă lim f(x)=+

x-> x 0

x-> x 0



n ∈ N, a>1

lim xn/ax=0 x-> x 0

lim ln(1+f(x))/f(x)=1

lim f(x)=0

x-> x 0

x-> x 0 f(x)

lim (a -1)/f(x)=ln a

daca lim f(x)=0

x-> x 0

x-> x 0 r

lim [(1+x) -1)]/x=r x-> x 0

lim sin f(x) / f(x)=1

daca lim f(x)=0

x-> x 0

x-> x 0 f(x)

lim (e -1)/f(x)=1

daca lim f(x)=0

x->0

x-> x 0

Cazuri de excepţie 0/0

- lim de funcţii raţionale in puncte finite a - lim de funcţii in compunere cu funcţia modul - sub radical de ordine diferite figurează aceeaşi expresie - sub radical figurează expresii diferite - lim trigonometrice

Se face simplificarea prin (x-a)k Se explicitează modulul Se schimba variabila, notându-se radicalul de ordin egal cu cel mai mic multiplu comun al ordinelor radicalilor cu alta variabila Se amplifica numărătorul si (sau) numitorul cu expresia conjugata lim sin f(x) / f(x)= lim tg f(x) / f(x)= lim arcsin f(x) / f(x)= lim arctg f(x) / f(x)=1 x-> x 0

∞ -∞ ∞

1

x-> x 0

- lim de funcţii raţionale

Se aduce la acelaşi numitor

- lim de funcţii iraţionale

Se amplifica cu conjugata lim (1+ f(x)) 1/f(x)= e

x-> x 0

x-> x 0

x-> x 0

0

0

lim x*ln x=0 si scrierea fg=e g* ln f x\>0

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