Theoremes de limites

Formes indeterminees

(+\infty) + (-\infty), \infty \times 0, \frac{0}{0}, \frac{\infty}{\infty}

Somme

Notation : lim f + lim g = lim (f+g)

L + L' = L+L'

+\infty + L' = +\infty -\infty + L' = -\infty +\infty + +\infty = +\infty -\infty + -\infty = -\infty

Produit

Notation : lim f \times lim g = lim (f \times g) L \times L' = L \times L' L \times \infty = \infty (*) \infty \times \infty = \infty (*) (*) du signe du produit

Quotients

Notation : \frac{lim f}{lim g} = lim \frac{f}{g}

\frac{L}{L' \different 0} = \frac{L}{L'} \frac{L \different 0}{0} = \infty (*) \frac{L}{\infty} = 0 (*) \frac{\infty}{L} = \infty (*) (*) du signe du quotient

Composition

\lim{a} u(v(x)) ? \lim{a} v(x) = c et \lim{c} u(x) = c' \implies \lim{a} u(v(x)) = c'

Suite geometrique

U_n = U_0 * q^n q \in [-1;1] \implies lim U_n = 0 q > 1 \implies lim U_n = \+infty

Comparaison

f(x) > g(x) et lim g(x) = +\infty \implies lim f(x) = +\infty f(x) < g(x) et lim g(x) = -\infty \implies lim f(x) = -\infty

Gendarmes

g(x) \le f(x) \le h(x); lim g(x)= lim h(x) = L \implies lim f(x) = L