203 bài tập về giới hạn hàm số
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143 BAI TAP GIOI HAN DAY SO - HAM SO - WWW.MATHVN.COM
I H N DÃY S
3
lim
6n − 2n + 1
lim
lim
n 3 − 2n
n 2 + 4n − 5
1 − n + 2n
4n 3 + 6n 2 + 9
2n 3
1 − 5n 2
lim
+
2n 2 + 3 5n + 1
lim
3
n3 + n
lim
n+2
(
)
3
lim
n + n + 3n + 2
lim
lim
3
4n
3n + 1
lim n
2 −1
2.3n + 4n
(
(
lim
3
n
1 + 2 + ... + n 3 3
1 + 2 + ... + n 3 =
2
11n + n + 2
lim
3
13 + 23 + ... + n 3
3n + n − 2
lim
n 6 − 7n 3 − 5n + 8
lim
n + 12
lim 1 + 2n − n
3n − 1 − 2n − 1
)
n2 + n + 2 − n + 1
(
lim (
lim
)
2
( n + 1)
2
2
+
3
3
1
1
1+ +
5
5
lim
3n − 2.5n
)
2
+ ... +
2
3
+ ... +
1
5
2
n
n
4 n − 5n
n
n +3 − n −5
lim 1 + 2 + ... + n
n2
3
lim n
2 + 3.5n
7 + 3.5n
n +1 − n
n2 + 1 − n + 1
lim
3n + 2
2n 2 + n + 1
lim
4
lim
)
1 − 3n 2
n. 1 + 3 + ... + (2n − 1)
1+
2
2n 2 − n
lim
n 6 + 5n 5
3
2
4
−2n 2 + n + 2
2n 2 − 3
lim
n 5 + n 4 − 3n
2n 2 − n + 3
lim
2
3
−3n 5 + 7n 3 − 11
4
lim
n 3 − 5n + 7
3n 4 + 5
7n 2 − 3n + 2
3n 3 + 2n − 1
lim
lim
n2 + 5
2n 2 − n
lim 2n − n + n + 2
n 2 + 4 + ... + 2n
3
2n 3 − 4n 2 + 3n + 3
2n 4 + 3n − 2
lim
lim 3n − 7n + 11
lim
5n 2 + n
n5 + n 4 − n − 2
lim
3n 3 + n 2 + 7
2
(
n2 + n + 1 − n
lim
(
lim
)
n2 − n + 3 − n
(−3)n + 5n
( −3)n +1 + 5n +1
(
lim n 2 n − n 2 + 1
)
lim
1
n + 2 − n +1
GI I H N HÀM S
( 3x 2
x→ 2
1. lim
)
5. lim x 2 − 4
+ 11) x
4x + 2
x −3
3x 6 − 2x 5 + 5
3
x →−2
2
x →−∞
x 6 − 5x + 1
10. lim
3x − x + 5
7. lim
x →9 9x − x 2
3
( 3x
3. lim
2
x →1
6. lim
x→ 3
9. lim
( 7x
+ 7x + 11 2. lim
x3 − 2
11. lim
3
+ 1)( 2 − 3x )
7x + 11
4. lim 2 x 1 −
x →0
x
x +1
8. lim
x →−∞
x2 + 5
2
2x 4 − 3x + 5
x 4 − 2x 2
12. lim
+
3− x
3− x
x →−∞
x →−∞ 6x − 3x + 2
x →3
3x − 2
5x − 2
3− x
3− x
x+2 x
4 − x2
x3 + 2 2
13. lim
14. lim
15. lim
16. lim
17. lim
x →3 3 − x
x → 0+ x − x
x → 2− 2 − x
x →3− 3 − x
x →− 2 x 2 − 2
x →+∞
18. lim
x 4 − 27x
x →3 2x 2
22. lim
x →+∞
1
19. lim
x
2x 4 + x 2 + 1
20. lim 3
2x 5 + x 3 − 1
21. lim
x 2 + x + 2x
2x + 3
( 2x 2 − 1)( x3 + x ) x →−∞
23. lim ( 2x 3 − 5x 2 + 3x − 1) 24. lim 2x 4 − 5x 2 + 1
x →+∞
x →+∞
x →−2 x 2
− 3x − 9
( x + 1)
x 4 − 16
+ 6x + 8
x →+∞
)
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2x + 1
x−2
25. lim
x → 2+
29. lim
x3 − 8
x →2 x 2
x → 2−
x →( −3)
x →4 x 2
x →3−
lim
x →−4 x 2
x →2
x →0
x →0
x →1 x 2
3− 5+ x
x →4 1 − 5 − x
lim
lim
x →2
7 + 2x − 5
x −3
x 2 + 2x − 15
x+5
lim
x →−∞
5+ x − 5− x
x
lim +
x →−2
x5 − x − 6
8 + 2x − 2
x+2
2
f (x) = mx
3
x →0
x− x+2
4x + 1 − 3
lim
x →+∞ 3x 2
− 5x − 2
x −1
lim
lim f (x)
3
lim
f ( x ) =
lim
lim
x →0
4 − x2 − 2
9 − x2 − 3
x 2 − 4x + 3
x →+∞
x 4 + 4x
x →−∞
x2 + 3 − 2
x 2 + 3x − 4
lim
x2 −1
1 − 3x + x 2 − 1 + x
x
x +1
x3 − 2
(x − 1)2
lim
x →+∞
x 4 − 5x 3 + 6
x 2 − 12x + 20
x 6 − 4x 4 + 4
x2 − x − 6
x →−∞
f (x) =
x →2
;x>2
lim
x →−∞ x 2
3
2x − 3x + 1
2x − 1 − x
x −1
x →1
x →−1
x →−∞
x2 − 1
x( x + 5) − 6
lim
+ 2x − 3
2 x − 3x
lim
+
x →0 3 x − 2x
;x≤2
lim
lim
lim
x2 − 4
x 2 − 25
x →1
lim
lim
x 2 + 3x − 10
x →0
x2 − x
x →1
x −1
x 2 + 4x
x + 4 −3
x →5
x−2 −2
x−6
x 2 − 3x + 2
x →−4
lim
1+ x − 1− x
x
x 2 + 3x − 4
3x − 5 − 1
x−2
x →2
lim
lim
x2 − x − 6
x →−2
lim
x →6
lim
x →1
x →+∞
(x − 1)2
x 3 + 4x 2 + 4x
lim
3x − 2 − 4x 2 − x − 2
lim
x 3 + 3x 2 + 2x
lim
x →−∞
x −3
2x + 10 − 4
lim
x →3
x →0
x →0
x 2 − 4x + 3
x →1
x3 − 1
x →1 x(x + 5) − 6
1 + x + x2 −1
x
lim
6x 2 + 3 + 3x
lim
+ 2x − 3
lim
x →9
x →5
lim
lim
5−x
5− x
lim
x +1
x →−1
− 5x − 2
lim
x →1 x 2 + 2x − 3
x+9 −2
x−7
lim
1 + x − x2 + x + 1
x
x →0
x −1
lim
lim
x →7
1 − 2x + x 2 − (1 + x )
x
lim
4
lim
x2 − 4
x →2 x − 2
x4 −1
x2 − x − 6
x →−2
x
1+ x −1
lim
lim
− 12x + 20
x2 + 5 − 3
.
x−2
lim
x2 + x +1 −1
3x
lim
x 2 + 2x − 15
lim
x →−5
x+5
x 3 + 3x 2 + 2x
9 − 3x 3
x →0
x 2 + 3x − 10
x → 2 3x 2
2x 2 + x + 10
37. lim
x2 − x
x2 +1
x →+∞
x →+∞
x −1
x3 − 5
lim
33. lim
x2 + x
x →1+
lim
x 2 + 2x − 15
lim
x →3
x −3
x 2 − 5x + 6
x3 + 1 − 1
36. lim
− 4x
x 2 − 2x
x → 2+
27 − x 3
x −1
lim
x →1 1 − x
)
x →0
x3 − 8
39. lim
(
2x 3 − 5x 2 + 3x − 1 28.
32. lim
x −2
35. lim
3− x
38. lim
x →+∞
( x + 3 )2
−
x2 − 3
x →− 3
lim
31.
−4
27. lim
2x 2 + 5x − 3
x3 + 3 3
34. lim
2x + 1
x−2
26. lim
3x − 1 ; x ≤ 1
2
x + 1 ; x > 1
x 2 − 5x + 6
mx + 4
;x>2
lim f (x)
x →1
Tìm m
hàm s
có gi i h
n
;x≤2
khi x → 2
lim x
x →+∞
2
lim
x →+∞
(
(
)
x2 + 1 − x2 − 2
lim
x →+∞
)
(
)
x 2 − 7x + 1 − x 2 − 3x + 2
(
x − 4 − x2 − 7 x + 2
x 2 − 2x + 1 − x 2 − 6x + 3
xlim
→+∞
)
lim
x →+∞
(
x 2 − 4x + 1 − x 2 − 9x
)
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60 BÀI T P GI I H N DÃY S
1, lim
n 2 - 2n + 1
3n 2 + n - 3
2, lim
4, lim
n n - n +1
n2 + 3
5, lim
2
8,
)
16, lim
4n 2 + n + 1
-n - 3
( 2n - 1)
n
3
19, lim
2007
- n + 3n - 1
n 3 - 4n + 1
-4n 3 + n 2 - 2
2007
- 3n
-1
17, lim
2000
+ 3n + 6 2n 2 - n - 1
3n - n + 1
2
( 2n
)(
+ 1)
(
4n + 1 2n 2 - 4n + 2
26, lim
)
( 5n
2
)(
29, lim
)
44, lim
4
2n + 3n - 2
)
)
)
n n - 2n + 6
3
n3 + 7
30, lim
( 2n
( 3n - 2 )
)
- 1 - ( n + 1)
)
n - 3 n +1
( n - 1)
lim
45, lim
3
n 2 - 3n + 1 - 2n
(
)
2
- n
3n + 2
)
(
51, lim (
)
n + 3 - n +1
n n + 3n - 1
5n + 7
n2 + 2 - n n
48, lim
)
n 2 + 4n + 2 - n + 2
54, lim n + 5
57, lim
2n - 5 - n + 2
n 2 + 3n + 3 n 3 + n 2 - 2n
(
3n + 2
2n + 1 - n
(
)
4
4n 2 - 3n + 1 - 2n
(
)
2
( -2 ) + 3 n
n
( -2 ) - 3n +1
(
42,
2
( 4n + 3)
53, lim n
59, lim
4n 2 - 1 + 2n - 1
39, lim
56, lim
)
(
)
n + 1 - n2 + 2
n 3 + 2n 2 + 1 - n
n + 2 n ( 3n - 1)
27, lim
(
(
50, lim (
2n 2 + 1 - 2n 2 + n + 1
3
5n - 3n + 2
36, lim - n 3 + 4n - 1
2
)
2
)
47, lim
1
(
3n 2 + 1 + n 2 + 2n - 1
(
3n 2 + 3n - 1
41, lim 3
2n - 2n + 1
+ n - 2n + 1
)
4
33, lim
2n 2 - n
40, lim
n +1
3
2
2 n +1 + 3n +1
2 n + 3n
38, lim 2n - n 3 + 1
58, lim
(
n
37, lim -2n 4 - 3 n + 1
55, lim
24, lim
2n + 4
35, lim n 2 - 3n - 10
n2 + 2 - n
1 + 2 + 3 + ... + n
n2
n
2
5 n - 3n
5 n +1 + 3n + 2
n2 + n + 5 - n
21, lim
n 2 - n + 3 ( 4n - 7 )
34, lim
(
49, lim (
52, lim (
n +1 - 2
18, lim
n +3
( 2n + 1) - ( 3n - 1)
32, lim
46, lim
)
3
4
2 n + 3n
3n + 1
( 2n - 1)
-2n n - 4n + 1
3
+ 3n - 1 2n + 6
31, lim
43, lim
n 3 + n 2 - 2n - 4n
15, lim
n 3 + 1 - n 2 + 2n
n 5 + 3n - 1
3
2
3
+ 2 -3n - 1
2
4
( n + 1 ) - ( n - 1)
lim
4
4
( n + 1 ) + ( n - 1)
2
2n 2 + 1 - n 2 + 1
n +1
23, lim
n
3n n - 2n + 1
n 3 + 3n + 1 - 3n 2 + 4
3n - 1
(
n + ( -1)
( n - 3)( -2n + 4n - 1)
12, lim
( 6n + 2n - 1) ( 2n - 1)
)
8n 4 + 4n 3 - 1
( 3n - 1) ( n
20, lim
2
28, lim
n +3
-6n - n + 1
n 1 + 3 + 5 + ... + ( 2n + 1)
3
9,
)(
2
( n + 1)( 2n - 5 )
( 3n - 1)( n + 2 )
4
n 2 + 1 - 3n - 1
14, lim
8n 3 + 2n - 1 + 3n
25, lim
6, lim
( n + 1) - 3n
lim
2
( 2n - 1)
2n - 4 n + 7
22, lim
3, lim
2
(n
11, lim
- 1 ( 3n + 2 )
- n 3 + 2n - 1
13, lim
( n + 1)( n + 2 )
2
4n + 6
7, lim
n -1
(n
10, lim
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)
)
n-2)
2n + 3 - 2n - 1
(
2n - 1 n +3
)
60, lim
(
3
n 3 + 3n 2 + 1 - n 2 + 2n
)
