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limiti-notevoli

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FORMULARIO
Limiti notevoli
1⎞
⎛
1) lim ⎜1 + ⎟
x⎠
x → ∞⎝
x
⎛ α⎞
2) lim ⎜1 + ⎟
x⎠
x → ∞⎝
=e
11) lim
x→0
= eα
12) lim
x→0
x
1
7)
lim
x → +∞ x β
=0
β
lim x ⋅ ln α x = 0; (∀α ∈ ℜ , ∀β > 0 )
x → 0+
1
ax −1
= ln a =
9) lim
log
x
e
x→0
a
8)
10 )
lim
x→0
(1 + x)α
x
−1
= α; α ∈ ℜ
Formule simboliche
non indeterminate
k
= ∞ K.............K k ≠ 0
0
∞ ⋅ k = ∞ KK..........k ≠ 0
k ⋅ (− ∞ ) = −∞ KK...k > 0
sin x
=1
13) lim
x→0 x
sin(ax)
= 1 (a ≠ 0)
14) lim
x → 0 ax
tan x
=1
15) lim
x→0 x
1 − cos x 1
=
16) lim
2
2
x→0 x
arctan x
=1
17) lim
x→0 x
arcsin x
=1
18) lim
x→0 x
sinh x
=1
19) lim
x→0 x
cosh x − 1 1
=
20) lim
2
x
x→0
3) lim (1 + x ) x = e
x→0
ln (1 + x )
=1
4) lim
x
x→0
ln (1 + α x )
=α
5) lim
x
x→0
ax
= +∞ , a > 1
6) lim
x → +∞ x β
ln α x
log (1 + x )
1
a
= log e =
; a ∈ ℜ + − {1}
a
x
ln a
k ⋅ (− ∞ ) = +∞ KK...k < 0
(+ ∞ ) ⋅ (− ∞ ) = (− ∞ ) ⋅ (+ ∞ ) = −∞
k −∞ = 0 KK.............k > 1
k −∞ = +∞ K....K 0 < k < 1
(+ ∞ )−∞ = 0
k
= 0 KK..............k ≠ ∞
∞
∞+k =∞
k ⋅ (+ ∞ ) = +∞ KK k > 0
k ⋅ (+ ∞ ) = −∞ KK k < 0
(+ ∞ ) ⋅ (+ ∞ ) = (− ∞ ) ⋅ (− ∞ ) = +∞
ex −1
=1
x
k +∞ = +∞ KK.......k > 1
k +∞ = 0 KK....0 < k < 1
(+ ∞ )+∞ = +∞
Limiti notevoli e limiti generalizzati
Limite notevole
sin x
=1
x
x→0
1 − cos x 1
=
lim
2
x2
x→0
log(1 + x)
=1
lim
x
x→0
lim
e x −1
=1
lim
x
x→0
⎛
1⎞
lim⎜⎝1+ x ⎟⎠
x
=e
x→∞
lim(1+ x)
1
x
=e
x→0
Formula generalizzata
sin f ( x )
=1
f (x )
x →0
lim x
lim
( )
f
log [1 + f ( x )]
=1
lim
f ( x)
f ( x )→ 0
e f ( x) − 1
=1
lim
f ( x)
f ( x )→0
⎛
f ( x )→ ∞
1 ⎞
⎟
f ( x ) ⎟⎠
lim [1 + f ( x )]
f ( x )→ 0
1
lim x
x → ±∞
a
1
lim x
x→0
a
x→ 0
lim cos x = 1
x→ 0
= ∞ KK .......... ........ a > 0
= +∞ KK .......... ........ a > 1
lim a
x
= 0 KK .......... ..... 0 < a < 1
lim a
x
= 0 KK .......... .......... a > 1
x → −∞
lim a = +∞ KK.......... .0 < a < 1
lim log x = +∞ K.......... K.a > 1
lim log x = −∞ KK...... 0 < a < 1
lim log x = −∞ K.......... K.a > 1
x
=e
x → −∞
a
x → +∞
lim sin x = 0
= 0 K .......... ......... K a > 0
x
x → +∞
=e
= +∞ KK .......... ....... a > 0
lim a
x → +∞
f ( x)
1
f (x)
a
x → +∞
1 − cos f ( x ) 1
=
lim
2
f 2 (x )
f ( x )→ 0
lim ⎜⎜⎝1 +
Limiti di Funzioni elementari
a
x → +∞
a
x→0+
lim log
x→0+
lim
x → +∞
n
a
x = +∞ K ......... 0 < a < 1
x = +∞
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