2 Limiti di funzioni Esercizi

2. Limiti di funzioni
2
4
Limiti di funzioni
Esercizi
A)
Calcolare i seguenti limiti:
1. lim
2x 3
3x 2 + 5x
x4 x
x!0
2. lim
2x 3
3x 2 + 5x
x4 x
2x 4
3x 3 + 5x
x6 x2
x!+1
3. lim
x! 1
4. lim
x3
4x 2 + 5x
x2 4
x!2
5. lim
x!1
6. lim+
x2
x3
2x 5
x!0
2
1
3x + 2
sin x + 5x 2
x2 x
sin(x + x 4 )
7. lim+
x!0
x3 x
sin(3x)
8. lim
x!0 tg(5x)
p
3+2 x
15. lim p
x!+1
x sin x
p
1+ x 1
16. lim
x!+1
e 3x 1
17. lim
x!0
sin(2x 2 )
cos2 x sin2 x
log(1 + 4x 2 )
x!0 sin(x 3 + 4x 4 )
p
p
19. lim
x4 + 1
x4 + x2 + 1
18. lim
x!+1
p
20. lim
x 2 + 2x 3 + x
sin2 x
x!0
21. lim+
p
x 2 + 2x 3 + x
sin2 x
x!0
22. lim
x e 3x
x!0
x2
cos(x + x 2 )
23. lim x 2 e 2/x
1
24. lim x 2 e 2/x
1
x!+1
x sin(3x) + x 4
9. lim
x!0 2x 2 + x sin(2x 3 )
p
x 2
10. lim
x!4 x sin(4
x)
sin(2x)
x!⇡/2 cos x
11. lim
sin x 2
x!0 sin x 2 + x
12. lim
13. lim+
p
x!0
p
p
x + x2
x + 2x 2
p
x x + 3x 2
4 + x2 2
14. lim p
x!0
1+ x 1
1 cos(3x)
x! 1
25. lim (x + 3)2 log
x!+1
1
2x 2
2x 2 + 1
cos(x 2 x 3 )
x!0 (x 2
2x 3 ) sin(x 2 3x 3 )
p
p
x sin 2x
27. lim
x!0 x + 3x 3/2
sin(x 2 )
p
exp x 5 + 3x 10
1
28. lim+
p
x!0 cos(3 + x 5 ) sin
2x 5
26. lim
1
x + x9
x + 2x 8
29. lim p
x!0
x 4 + 8x 6 sin x 5
log
2. Limiti di funzioni
30. lim
x!0
p
5
x 4 + 2x 6 sin(5x 2 )
39. lim
log cos(4x 2 + 2x 3 )
cos(6x 2 )
31. lim
x!0 x 2
log(x 2
40. lim
log 3
x!+1
✓
◆
3 + 7x 2
exp
e3
1 + 3x
32. lim
Å
ã
Å
ã
3 + 7x
3 + 7x
x!0
log
cos
3 + 3x 2
3 + 3x 2
2x + x 2
p
x 4 + x 2 e 1/(x+3)
1
2x + 1
◆
x4 + 3
41. lim log
exp(1 + 5 log x 2 ) x
x!+1
x4 + 1
✓
6
◆6
5 + x2
5 + x2
log
log
5 + 4x
1 + 4x
p
p
5x 10 + x 11
5 x5
e 2x x 3 + 1
x!+1 e x + 3x 4
42. lim+
e x+1 + e x/2 x 2
x!+1
e x 1 e x/2
3 + x5
1 + 3x 2
x5
43. lim log
log 1 + e sin
x!+1
x + x5
1 + 5x 3
x!0
34. lim
✓
e x+1 + e x/2 x 2
1
e x 1 e x/2
5x 2 + 2
exp
2x 2 + 1
35. lim
x!
log p
36. lim
x!+1
sin p
38. lim
x!+1
x4
x!0
x 4 + 2x 3
x
3x 4
+
45. lim
+ 2x 3
x3
x
2
⇣
8
+x
16
ä
⇣ x ⌘
exp
x +2
p
e
1
cos(3x 2 ) 1
✓
Determinare a 2 R⇤ e b 2 R tali che:
p
1. 2 log 3 + x
4 + 4x ⇠ a x b
x !0
p
2 + 6x 2
exp
p
2
✓
6x 2 ⇠ ax b
◆
✓ 2
◆
x2 + 3
3x + 1 p 6
3. log
log
4x + 1 ⇠ ax b
x2 + 4
4x 2 + 1
✓
◆
ã
Å
ã✓
x
x
1
exp p
4. sin
cos
x2 + 6
x2 + 3
6x 2 + 1
Å
e 3 cos x
e3
2
log(1 + 3x) sin(1 + 5x)
p
x6
6x 3
⇠ axb
x3 + 3 ⇠ a x b
x !0
x ! +1
e
sin(3x 2 )
◆
◆
x + 3x 2
2
exp
e sin e 6x 1
3
x + 6x
46. lim
p
x!0
x x 4 + x 8 arctan(1 + 3x)
B)
2. exp
e2
x 4 + x 6 exp(1 + x 2 + x 4 )
✓
⌘
◆
sin2 log(1 + 5x) cos2 log(1 + 5x)
x!0
p
exp 3 x
x!+1
p
44. lim
x2
Ä
37. lim x 4 log 2
6.
1 sin
✓
33. lim
5.
2
4e x + x
log cos(3x)
x!0
1
+ 3)
p
2 4
x !0
x! 1
◆
1 ⇠ ax b
x ! +1
2. Limiti di funzioni
7.
p
6
✓
x 8 + 2x 2
x
4
x
2
◆
✓ 2
◆
x5 + 4
4x + 4
log
exp
⇠ ax b
5
2
x + 2x + 4
x +2
p
x 4 + 10x 3 + 2x 2 x 2 5x 2 ⇠ ax b
x ! +1
p
p
9. (1 + x 2 ) x 6 + 6x 4 + x 2
x6 + x2
3x 3 ⇠ ax b
x ! 0+
8.
10. (1 + x 2 )
p
x 6 + 6x 4 + x 2
p
x6 + x2
x 6 + 6x 4 + x 2
p
x !0
11. (1 + x 2 )
p
3x 3 ⇠ ax b
x6 + x2
x 6 + 6x 4 + x 2
p
x ! +1
12. (1 + x 2 )
p
3x 3 ⇠ ax b
x6 + x2
3x 3 ⇠ ax b
x! 1
Determinare a, b 2 R tali che:
Å ã
1
1. (x + 3) exp
= a x + b + o(1)
x
Å
ã
1
2. (x + 3) exp
= a x + b + o(1)
x 2
p
3. x 4 + 6x 3 x 2 = a x + b + o(1)
C)
x ! +1
x ! +1
x ! +1
2
(x 2 + 2)e 1/x
4.
= a x + b + o(1)
x +3
x ! +1
5.
(x + 2)(x 2 + 1)
= a x + b + o(1)
p
x x 2 + 4x + 5
x ! +1
6.
(x + 2)(x 2 + 1)
= a x + b + o(1)
p
x x 2 + 4x + 5
x! 1
7.
8.
p
p
x 2 + 3x + 3
1 + e 2/x
x
x 2 + 3x + 3
1 + e 2/x
x
= a x + b + o(1)
x ! +1
= a x + b + o(1)
x! 1
x ! +1
2. Limiti di funzioni
7
Soluzioni
A)
1.
5
26.
2. 0
27.
3. 0
1
4.
4
5.
6. 1
7.
2
28. p
1
1
3
8.
5
3
9.
2
10.
2 cos 3
29.
2
30.
5
8
31.
54
32.
27e 3
7 cos 1
1
16
34. e 2
1
2
37.
3
38.
e
39.
29
72
40.
1
2
12. 0
13.
1
33. +1
11. 2
14. 0
15. 2
16. 0
17. 2
18.
1
2
p
1
35. +1
p
36.
3
41. 2e
19.
1
2
42.
p
2 5 46 log 5
56
20.
1
43.
3
5
21. +1
22.
6
23. +1
24.
25.
1
1
2
44.
e2
25
45.
2e
9
46.
36e
⇡
2. Limiti di funzioni
8
B)
1
b =2
4
p p
2. a = 3 2 e 2 b = 2
1. a =
3. a = 2 log
3
b =1
4
1
4. a = p b = 2
6
5. a =
6. a =
3e 6
b =3
4
9
b= 3
2
7. a = e 4 b = 12
8. a =
27
b =0
2
9. a =
3
b =5
2
10. a = 6 b = 3
11. a =
3
b =1
2
12. a = 6 b = 3
C)
1. a = 1 b = 3
5. a = 1 b = 0
2. a = 1 b = 2
6. a = 1 b = 0
3. a = 3 b =
9
2
4. a = 1 b = 3
7. a = 0 b =
3
4
8. a = 1 b =
1
4