1. 1 Ejercicios de límites Ficha I
Calcula los siguientes límites
x2 5x + 2 1
1. lim 2
1
x! 2 x + 3x 3 5
2x2 7x 4
2. lim 0
x!4 x3 + x2 10
x+2
3. lim +1
x!3+ x2 4x + 3
x+2
4. lim 1
x!3 x2 4x + 3
x+2
5. lim No existe (1 )
x!3 x2 4x + 3
x2 + 5
6. lim + 1:(2 )
x!3 x2 6x + 9
x+1
7. lim +1
x! 1 x3 + 3x2 + 3x + 1
x2 + 4x + 4
8. lim . No existe
x! 2 x3 + 6x2 + 12x + 8
x + 4x + 4 = (x + 2)2
2
Ayuda:
x + 6x2 + 12x + 8 = (x + 2)3
3
r r
x3 2x + 5 26
9. lim
x!3 4x2 x + 2 35
x2 25
10. lim 5
x!5 10 2x
(x 1)(x 3)3
11. lim 0
x!3 (2x + 1)(2x 1)(x 3)2
x2 16
12. lim p 24
x!4 2x + 1 3
p
x+1 2 1
p
13. lim p p 3 3
x!1 2x + 10 5x + 1
p
x+1 2 2
14. lim p p
x!3 2x + 10 5x + 1 3
1 x+2 x2 + 5
lim = +1 y lim 2 = 1
x!3+ x2 4x + 3 x!3 x 6x + 9
2 x2 6x + 9 = (x 3) 2
1

2. x5 + a5 5
15. lim a
x! a x4 a4 4
(x + h)4 x4
16. lim 4x3
h!0 h
x2 4 1
17. lim
x!2 x3 2x2 + 4x 8 2
3 2
2x + 9x + 12x + 4 3
18. lim
x! 2 3x3 + 11x2 + 8x 4 7
p
x+4 2 1
19. lim
x!0 x 4
p p
x+h x 1
20. lim p
h!0 h 2 x
p3
r
x2 1 6 4
21. lim p
6
x!1 x3 3x + 2 3
2x 10
22. lim p 12
x!5 x + 4 3
x2 + 2x 8
23. lim 6
x!2 x3 6x2 + 11x 6
x2 x 1
24. lim
x!0 5x2 + 3x 3
x3 4x2 + 4x 2
25. lim
x!2 x3 3x2 + 4 3
x 1
x2 13x + 22 2 3
26. lim
x!2 x2 8x + 12 2
px
13 1
27. lim
x!3 x3 6
p
x 6+x 5
28. lim
x!3 x 3 6
x4 1 4
29. lim
x! 1 x3 + 1 3
x2 + x 2
30. lim p 12
x!1 x + 3 2
2x 4
p
31. lim
x!3
ex 2 4
e
2

3. p
3
x+6 2 1
32. lim 12
x!2 x2 + 3x 2
0
Este límite presenta la indeterminación 0
p 3
A3 B 3 p 3
x+6 23
Ayuda: Como A B = 2 + AB + B 2
entonces: 3 x + 6 2 = p 2 p
A 3
x+6 +23x+6+4
p p 3
3
x+6 2 3
x+6 23
lim = lim h p p i =
x!2 x2 + 3x 2
2 x!2 3 x + 6 + 2 3 x + 6 + 4 ( x2 + 3x 2)
x 2
lim h p 2 p i =
x!2 3 3
x+6 + 2 x + 6 + 4 ( x + 1) (x 2)
1 1 1
lim h p 2 p i = (4+4+4)( 1) = 12
x!2 3 3
x+6 + 2 x + 6 + 4 ( x + 1)
p
3
3x 2 + 2
33. lim
x! 2 x2 + 3x + 10
0
Este límite presenta la indeterminación 0
p 3
A3 + B 3 p 3
3x 2 + 23
Al ser A+B = 2 2
entonces: 3 3x 2+2 = p 2 p
A AB + B 3
3x 2 2 3
3x 2 +4
p p 3
3
3x 2 + 2 3
3x 2 + 23
lim = lim p p =
x! 2 x2 + 3x + 10 x! 2 3
3x 2
2 3
2 3x 2 + 4 ( x2 + 3x + 10)
3 (x + 2)
lim h p 2 p i =
x! 2 3 3
3x 2 2 3x 2 + 4 (x + 2) ( x + 5)
3 3 1
lim h p 2 p i = (4+4+4)7 = 28
x! 2 3 3
3x 2 2 3x 2 + 4 ( x + 5)
p
x 1 3
1
34. lim 6
x!1 x2 + 1
p3
x+1 1
35. lim 6
x! 1 x2 + 1
p
5
7x 3 2 7
36. lim 800
x!5 x2 + 25
A5 B 5
Ayuda: A B=
A4 + A3 B + A2 B 2 + AB 3 + B 4
p
5
3 7x + 3 + 2
37. lim
x! 5 x2 + 25
3A A5 + B 5
+B =
A4 A3 B + A2 B 2 AB 3 + B 4
3