Limiti notevoli
Limiti di funzioni
Forma standard Forma con gli “o” piccolo
lim
x→0
sin x
x= 1 sin x=x+o(x), x →0
lim
x→0
1−cos x
x2=1
2cos x= 1 −1
2x2+o(x2), x →0
lim
x→0
loga(1 + x)
x=1
log aloga(1 + x) = 1
log ax+o(x), x →0, ∀a > 0, a= 1
lim
x→0
log (1 + x)
x= 1 log (1 + x) = x+o(x), x →0
lim
x→0
ax−1
x= log a ax= 1 + xlog a+o(x), x →0, ∀a > 0
lim
x→0
ex−1
x= 1 ex= 1 + x+o(x), x →0
lim
x→0
(1 + x)α−1
x=α(1 + x)α= 1 + αx +o(x), x →0, ∀α∈R
lim
x→+∞
xk
ax= 0 xk=o(ax), x →+∞,∀a > 1, ∀k∈R
lim
x→−∞ |x|kax= lim
x→−∞
ax
1
|x|k
= 0 ax=o(1
|x|k), x → −∞,∀a > 1, ∀k > 0
lim
x→−∞ |x|k
ax= 0 |x|k=o(ax), x → −∞,∀0< a < 1, ∀k∈R
lim
x→+∞xkax= lim
x→+∞
ax
1
xk
= 0 ax=o(1
xk), x →+∞,∀0< a < 1, ∀k > 0
lim
x→+∞
logax
xk= 0 logax=o(xk), x →+∞,∀a, k > 0, a= 1
lim
x→0+xklogax= lim
x→0+
logax
1
xk
= 0 logax=o(1
xk), x →0+,∀a, k > 0, a= 1
lim
x→±∞ (1 + 1
x)x
=e
lim
x→±∞ (1 + a
x)x
=ea,∀a∈R
lim
x→0(1 + x)1
x=e
lim
x→0(1 + ax)1
x=ea,∀a∈R
c
2017 Politecnico di Torino
