Tavola dei limiti notevoli
Fabio Durastante
December 14, 2009
Limiti per x→0di f:R→R:
lim
x→0(1 + x)1
x=e(1)
lim
x→0
(1 + x)α−1
x=α α 6= 0 (2)
lim
x→0+xx= 1 (3)
lim
x→0+x1
x= 0 (4)
lim
x→0
ex−1
x= 1 (5)
lim
x→0
ax−1
x= log a(6)
Funzioni logaritmiche:
lim
x→0+xblog x= 0 ∀b > 0 (7)
lim
x→0
log(1 + x)
x= 1 (8)
lim
x→0
loga(1 + x)
x=1
log a(9)
Funzioni trigonometriche:
lim
x→0
sin x
x= 1 (10)
lim
x→0
1−cos x
x2=1
2(11)
lim
x→0
1−cos x
x= 0 (12)
lim
x→0
tan x
x= 1 (13)
lim
x→0
arcsin x
x= 1 (14)
lim
x→0
arctan x
x= 1 (15)
Limiti per x→ ∞ di f:R→R:
lim
x→±∞ 1 + 1
xx
=e(16)
lim
x→+∞
ax
xb= +∞ ∀a > 1, b > 0 (17)
lim
x→−∞ ax|x|b= 0 ∀a > 1, b > 0 (18)
lim
x→+∞
x
√x= lim
x→+∞x1
x= 1 (19)
lim
x→∞
ex
xb=∞(20)
lim
x→+∞
log x
xb= 0 ∀b > 0 (21)
lim
x→+∞
log x
ex= 0 (22)
Forme indeterminate:
0
0,∞
∞,0·∞,1∞,00,(±∞)0,+∞−∞.
Confronto di infiniti e infinitesimi:
Se
lim
n→∞ |an|=∞
Allora:
logan≤nb≤cn≤n!≤nncon a, b, c > 1
Se
lim
n→∞ |an|= 0
Allora:
1
logan≥1
nb≥1
cn≥1
n!≥1
nncon a, b, c > 1
1
