Đề cương ôn tập môn Toán lớp 11 _Tập 8

PHẦN 1. GIỚI HẠN DÃY SỐ
lim
1.

6n 3 - 2n + 1
n 3 - 2n

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lim
2.

2n 3 - 4n 2 + 3n + 3
n 3 - 5n + 7

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lim
3.

- 2n 2 + n + 2
3n 4 + 5

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lim
4.

n5 + n4 - n - 2
4n 3 + 6n 2 + 9

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5.

æ 2n 3
1 - 5n 2 ử
ữ
ữ
lim ỗ 2
+
ỗ
ữ
ỗ2n + 3
ữ
ỗ
5n + 1 ø
è

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Page 1 of 23


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lim
6.

- 3n 5 + 7n 3 - 11
n 5 + n 4 - 3n

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2n 2 - n
1 - 3n 2

lim
7.

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3

lim
8.

n3 + n
n+2

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lim
9.

2n 4 + 3n - 2
2n 2 - n + 3

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3

lim
10.

n 6 - 7n 3 - 5n + 8
n + 12

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Page 2 of 23


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n2 + 1 - n + 1
3n + 2

lim
11.

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(

)

lim 3n 3 - 7n + 11

12.

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13.

lim 2n 4 - n 2 + n + 2

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14.

lim 3 1 + 2n - n 3

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Page 3 of 23


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lim
15.

1 + 2 + ... + n
n2

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2

n

ổử
ỗ2 ữ + ... +
ỗ ữ
ữ
ỗ3 ứ
ố ữ

ổử
ỗ2 ữ
ỗ ữ
ỗ ữ
ữ
ố3 ứ

1
1+ +
5

lim

2
1+ +
3

2
ổử
1ữ
ỗ ữ + ... +
ỗ ữ
ỗ5 ứ
ố ữ

n
ổử
1ữ
ỗ ữ
ỗ ữ

ỗ ữ
ố5 ø

16.
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17.

4n
lim
2.3n + 4n

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lim
18.

3n - 2.5n
7 + 3.5n

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Page 4 of 23


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lim

(

3n - 1 -

2n - 1

)

19.
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lim

(

n2 + n + 1 - n

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20.
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lim

(

n2 - n + 3 - n

)

21.
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Page 5 of 23


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lim
22.

1
n+ 2-

n+1

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PHẦN 2. GIỚI HẠN HÀM SỐ

(

)

lim 3x 2 + 7x + 11
1.

x ®2

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lim

(

x ®1

2.

7x + 11) x
4x 2 + 2

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lim

x đ- Ơ

3.

3x 2 - x + 5
x3 - 2

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Page 6 of 23


4.

2x 4 - 3x + 5
lim
x đ- Ơ
x 4 - 2x 2

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lim

x đ- Ơ

5.

x 6 - 5x + 1
5x 3 - 2

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lim
+

6.

x ®3

3- x
3- x

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x 4 - 27x
x ® 3 2x 2 - 3x - 9

lim

7.

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8.

x 4 - 16
lim
x ®- 2 x 2 + 6x + 8
Page 7 of 23


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x 2 + x + 2x
2x + 3

lim

x đ- Ơ

9.

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(

)

lim 2x 3 - 5x 2 + 3x - 1

10.

x ®+ ¥

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lim
+

11.

x ®2

2x + 1

x- 2

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Page 8 of 23


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lim
x ®2

12.

x3 - 8
x2 - 4

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lim
x ®0

13.

x3 + 1 - 1
x2 + x

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lim
x ®2

14.

x 2 + 3x - 10
3x 2 - 5x - 2

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x 2 + 3x - 4
x ®- 4
x 2 + 4x
lim

15.

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Page 9 of 23


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lim
x ®2

16.

x2 + 5 - 3
.
x- 2

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lim
x ®5

17.

x+ 4- 3
x 2 - 25

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lim
x ®3

18.

x- 3
2x + 10 - 4

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Page 10 of 23

5+ x x

lim
x ®0

19

5- x

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lim

3-

5+ x

1-

5- x

x ®4

20.

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3

lim

x ®- 1

21.

x+1

x2 + 3 - 2

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Page 11 of 23


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lim

x-

x ®2

x+ 2

4x + 1 - 3

22.

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3

lim
x®0

23.

1+ x - 3 1- x
x

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3

lim
x ®0

24.

1 + x2 - 1
x2

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Page 12 of 23


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3

lim
x ®1

25.

x+ 7- x+ 3
x 2 - 3x + 2

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2 1+ x x®0
x

lim
26.

3

8- x

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Page 13 of 23


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lim
x®0

26.

1 + 2x - 3 1 + 3x
x2

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PHẦN 3.HÀM SỐ LIÊN TỤC

1. f(x) =

ì 4 - x2

ï
ï
khi x < 2
ï
í x- 2
ï
ï 1 - 2x khix > 2
ï
ï
ỵ

tại xo = 2

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2. f(x) =

ì x3 - x - 6
ï
ï
ï 2
ïx - x- 2
ớ
ù 11
ù
ù
ù 3
ù
ợ

khi x ạ 2
khi x = 2
ti xo = 2

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3. f(x) =

ì 1 - 2x - 3
ï
ï
ï
khi x ạ 2
ớ
2- x
ù
ù 1 khi x = 2
ù
ù
ợ

ti xo = 2

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4. f(x) =

ì x 2 - 3x + 2
ï
ï
khi x ³ 1
ï
ï
x2 - 1
í
ï x
ïkhi x < 1
ï
ï 2
ï
ỵ

tại xo = 1

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5. f(x) =

ì
ï
ï x + 3 khi x £ 0
ï
ï
2
ï
í
ï x+ 1- 1
ï
khi x ³ 0
ï3
ï 1+ x - 1
ï
ỵ

tại xo = 0

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6.

ì x 2 - 3x + 2
ï
ï
(x > 1)
ï
ï
x- 1
f(x) = í
ï x
ï(x £ 1)
ï
ï 2

ï
ỵ
trên tập xác định

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7.

ì 4 - x2
ï
ï
(x ¹ 2)
b) f(x) = ï x - 2

í
ï
ï- 3
(x = 2)
ï
ï
ỵ
trên tập xác định

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8. Tìm A để hàm số

sau có giới hạn tại x0 = 1 với

ì x3 - 1
ï
ï
(x < 1)
f(x) = ï x - 1

í
ï
ï Ax + 2
(x £ 1)
ï
ï
ỵ

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9. Xét

tính liên tục của các hàm số

ì x3 + x + 2
ï
ï
ï
3
ï
f(x) = ï x + 1
ớ

ù4
ù
ù
ù3
ù
ợ

khi x ạ - 1
khi x = - 1
trên tập xác định

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10. Tìm m để hàm số

ì x 3 - x 2 + 2x - 2
ï
ï
khi x ¹ 1
f(x) = ï
í
x- 1
ï
ï 3x + m
khi x = 1
ï
ï
ỵ

x0 = 1
liên tục tại

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PHẦN 4. CHỨNG MINH PHƯƠNG TRÌNH CĨ NGHIỆM
1. Chứng minh rằng phương trình:

x 5 - 5x 3 + 4x - 1 = 0

có 5 nghiệm trên (–2; 2).

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2. Chứng minh phương trình x3

– 3x2 + 3 = 0 có 3 nghiệm phân biệt

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3. Chứng minh phương trình x3 + 3x2 – 3 = 0 có 3 nghiệm trong khoảng (– 3;1)
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4. Cho 3 số a,b,c khác nhau .CMR phương trình (x – a)(x – b) + (x – b)(x – c) + (x – c)(x – a) = 0 có 2
nghiệm phân biệt
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5.

Cho phương trình

x0 >

7

x4 – x – 3 = 0. Chứng minh rằng: phương trình có nghiệm x o ∈ (1;2) và

12

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6. Chứng minh phương trình (m2 + m + 1)x4 + 2x – 2 = 0 ln có nghiệm với mọi m
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7. Chứng minh phương trình

x 3 + ax 2 + bx + c = 0

ln có nghiệm

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Page 21 of 23


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ax 2 + bx + c = 0

8. Cho a ≠ 0 và 2a + 6b + 19c = 0. Chứng minh rằng phương trình:
ln có
é 1ù
ê ú
0;
ê 3ú
ë û
nghiệm x ∈
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a
b
c
+
+

=0
m+2 m+1 m

9. Cho m > 0 và a, b, c là 3 số thực thoả mãn:
. Chứng minh rằng
2
f(x) = ax + bx + c = 0
phương trình:
có ít nhất một nghiệm thuộc khoảng (0; 1).
ỉ + 1÷
ư
m
c2
÷= f(0).f ỗ
< 0
ỗ
ỗm + 2 ữ
ố
ứ
m(m + 2)
HD: Xột 2 trường hợp c = 0; c ≠ 0. Với c ≠ 0 thì
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Page 22 of 23


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