limiti notevoli

LIMITI NOTEVOLI
1.
2.
3.
+ ∞ → a > 1
lim a =
→+∞ 0 → 0 < a < 1
x
x
0 → a > 1
lim a =
→−∞ + ∞ → 0 < a < 1
x
x
+ ∞ → b > 0
lim x =
→+∞ 0 → b < 0
b
x
4. lim log(
x)
= +∞
x→+∞
5. lim log(
x)
= −∞
6.
+
x→0
log
x
lim
x→+∞
b
( x)
= 0
( b > 0)
7. lim x log(
x)
= 0 ( b > 0)
b
8.
+
x→0
lim
x→+∞
lim
x→−∞
x
a
a
x
b
x
x
b
= 0
= 0
x
9. lim x ⋅ e = 0
x→−∞
1
10. lim 1+
= e
x→±∞
x
11.
lim
sen
x→0
x
x
( x)
= 1
( a > 1,
b > 0)
x x0
12. lim a = a ( a > 0)
x→x
0
b
13. lim x = x ( x > 0)
b
x→x
0
0
14. log(
x)
= log(
x ) ( x > 0)
lim
x→x0
15. sen(
x)
= sen(
x )
lim
x→x0
16. cos(
x)
= cos(
x )
17.
18.
lim
x→x0
log
a
lim
x→+∞
x
x→±∞
( x)
= 0
b
lim 1+
= e
x
x
b
0
0
0
0
( a > 1)
0
19.
log
x
lim
x→0
b
( x)
x
a −1
x
= 0
20. lim = log(
a)
x→0
21. lim tg(
x)
π
x→−
2
22. lim tg(
x)
x→ 2
π
−
+
= −∞
= +∞
23. lim arctg(
x)
x→−∞
24. lim arctg(
x)
x→+∞
=
=
π
−
2
π
2
25. lim ctg(
x)
= +∞
+
x→
0
26. lim ctg(
x)
= −∞
−
x→
π
27. lim arcctg(
x)
= π
x→−∞
28. lim arcctg(
x)
= 0
x→+∞
n 29. lim n = 1
n→+∞
n
30. lim = e
n→+∞
n n!
31.
32.
33.
lim
→ 0
+
x
lim
x
−log
log
( x)
= 0
( 1+
x)
( )
+
x→0
log x
lim
+
x→0
xlog
x = 0
= 0
34. lim x − log(
1 + t)
x→0
35. lim
( x,
y ) →(
0,
0)
36. lim y
( x,
y ) →(
0,
0)
37.
lim
x→x0
f
g
( x)
( x)
x log
x
( y)
( b > 0)
= 0
= non ∃
= non ∃
⎯⎯
⎯ →0
ord. =
⎯ ⎯⎯ →0
ord. β
α o
l →α
= β
→α
> β
∞ →α
> β