Formulario Analisi matematica 1

Tavole e formulari

Formule notevoli

22 ∀x ∈ cos x + sin x = 1, R π ∀k ∈ + kπ sin x = 0 se x = kπ, cos x = 0 se x = Z 2π sin x = 1 se x = + 2kπ, cos x = 1 se x = 2kπ

2 π −1 − −1 sin x = se x = + 2kπ, cos x = se x = π + 2kπ 2

± sin(α β) = sin α cos β cos α sin β ± ∓ cos(α β) = cos α cos β sin α sin β

2 − sin 2x = 2 sin x cos x, cos 2x = 2 cos x 1−

x y x + y− sin x sin y = 2 sin cos 2 2− x + y x y− −2 cos x cos y = sin sin 2 2−

− sin(x + π) = sin x, cos(x + π) = cos x π π − sin(x + ) = sin x) = cos x, cos(x + 2

x a y x+y x y x−y x x y a = a a, a =, (a ) = a y a

∀x, log (xy) = log x + log y, y > 0 a a a x − ∀x, = log x log y, y > 0 log a a a y y

∀x ∀y ∈ log (x ) = y log x, > 0, R a a

Limiti notevoli

498 Tavole e formulari

α α lim x = +∞, lim x = 0, α > 0 x→+∞ +x→0

α α lim x = 0, lim x = +∞, α < 0 x→+∞ +x→0

n a x + . . . + a x + a a n 1 0 n n−m lim = lim xm b x + . . . + b x + b b x→±∞ x→±∞ m 1 0 m x x

lim a = +∞, lim a = 0, a> 1 x→+∞ x→−∞

lim a = 0, lim a = +∞, a< 1 x→+∞ x→−∞

−∞ lim log x = +∞, lim log x =, a> 1 a a x→+∞ +x→0

−∞ lim log x =, lim log x = +∞, a< 1 a a x→+∞ +x→0

lim sin x, lim cos x, lim tan x non esistono x→±∞ x→±∞ x→±∞

π ∓∞ ∀k ∈ ± lim tan x =, , lim arctan x = Z 2 x→±∞ ±( )π x→ +kπ 2

π ± lim arcsin x = = arcsin(±1) 2 x→±1

lim arccos x = 0 = arccos 1, lim arccos x = π = arccos(−1) x→+1 x→−1

− 1 1 cos x sin x = 1, lim = lim 2x x 2 x→0 x→0

"! x a 1 a ∈ = e, a, lim (1 + x) lim 1 + = e R x x x→±∞ x→0

log (1 + x) 1 log(1 + x) a lim =, a > 0; in parti