Tài liệu Tìm giới hạn (Bài tập và hướng dẫn giải) pptx

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Hà Nội, ngày 10 tháng 06 năm 2010
BTVN NGÀY 10-06
Tính các giới hạn sau đây.

( )
3
2
x 2
x 0
2
x 0
2
x 0
x
sin x sin x
2
x 0
x
x
4 x
1 Bµi1: lim
x
cos
4
sin sin sinx
2 Bµi 2 : lim
x
1 cosx cos2x
3 Bµi 3: lim

x
1 cosx cos 2x cos 2010x
4 Bµi 4 : lim
x
ln sin x cosx
5 Bµi 5 : lim
x
e cos2x
6 Bµi 6 : lim
x
x 3
7 Bµi 7 : lim
x 1
8 Bµi8
→
→
→
→
→∞
−
→
→+∞
−
−
π
−
−
−
−
−

+
−
−
−
+
 
−
 ÷
+
 
−
(
)
3
3 2 2
x
3
x
2
2
x 0
3
x 0
3 2
x
: lim x 3x x x 1
tan x sin x
9 Bµi 9 : lim
x
1 x cos x

10 Bµi10 : lim
x
1 tan x 1 sin x
11 Bµi11: lim
x
x x 2
12 Bµi12 : lim
sin(x 1)
→+∞
→∞
→
→
→∞
+ − − +
−
−
+ −
−
+ − +
−
+ −
−
−
………………….Hết…………………
BT Viên môn Toán hocmai.vn
Trịnh Hào Quang
Hocmai.vn – Ngôi trường chung của học trò Việt 1
TRUNG TÂM HOCMAI.ONLINE
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Hà Nội, ngày 28 tháng 02 năm 2010

HDG CÁC BTVN
• BTVN NGÀY 09-06:
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( )
( ) ( )
( )
( ) ( ) ( ) ( ) ( ) ( )
( )
x 0
x 0
x 0
x 0 x 0
m
n
x 1
m 1 m 2
x 1
1 x 1 2x 1 3x 1
*Bµi1: lim
x
1 x 1 2x 1 3x 1 x 1 2x 1 x 1 2x 1 x 1 x 1
lim
x
1 x 1 2x 1 3x 1 1 x 1 2x 1 x
lim
x
3x 1 x 1 2x 2x 1 x x 3 1 x 1 2x 2 1 x 1
lim lim 1 2 3 6

x 1
x 1
*Bµi 2 : lim
x 1
x 1 x x
lim
→
→
→
→ →
→
− −
→
+ + + −
+ + + − + + + + + − + + + −
=
+ + + − + + + − +
=
+ + + + + + + + + +
= = = + + =
−
−
− + +
=
( )
( )
( )
( )
( )
( )

( )
( )
( )
( )
( )
m 1 m 2
n 1 n 2 n 1 n 2
x 1
100
50
x 1
100 99 98
50 49 48
x 1 x 1
x 1 x x x 1
m
lim
n
x 1 x x x 1 x x x 1
x 2x 1
*Bµi3 : lim
x 2x 1
x 1 2(x 1) x 1 x x x 1 2
98 49
lim lim
48 24
x 1 2(x 1) x 1 x x x 1 2
− −
− − − −
→

→
→ →
+ + + + + +
= =
− + + + + + + + +
− +
− +
− − − − + + + + −
= = = =
− − − − + + + + −
( )
( )
( )
( )
( )
( )
( )
( )
20
2
10
x 2
3
20 20 20
10
20 20 20
10 10
20 10
x 2 x 2 x 2
2 2

x 0
x 0 x 0
x x 2
*Bµi 4 : lim
x 12x 16
x 2 (x 1) x 2 (x 1) x 2 (x 1)
3
lim lim lim
(x 2) (x 4) 2
(x 2) (x 4) (x 2) (x 4)
x 9 x 16 7
*Bµi 5: lim
x
x 9 3 x 16 4 x
lim lim lim
x
x x 9 3
→
→ → →
→
→ →
− −
− +
− + − + − +
 
= = = =
 ÷
− +
 
− + − +

+ + + −
+ − + + −
= = +
+ +
( )
( ) ( )
x 0
x 0 x 0
x
x x 16 4
1 1 1 1 7
lim lim
6 8 24
x 9 3 x 16 4
→
→ →
+ +
= + = + =
+ + + +

Page 2 of 8
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Hà Nội, ngày 28 tháng 02 năm 2010
( ) ( ) ( ) ( )
( )
( )
( ) ( )
( )
3

x 0
3 3
x 0 x 0 x 0
x 0 x 0
2
3
3
3
2
x 0
3
2
2
x 0 x 0
2 1 x 8 x
*Bµi 6 : lim
x
2 1 x 1 8 x 2 2 1 x 1 8 x 2
lim lim lim
x x x
2x x 1 13
lim lim 1
12 12
x 1 x 1
x 8 x 2 8 x 4
2x 1 1 3x
*Bµi 7 : lim
x
2x 1 (x 1) 1 3x (x 1)
2x 1 (x 1)

lim lim
x
x
→
→ → →
→ →
→
→ →
+ − −
+ − − − − + − − −
= = −
−
= − = + =
 
+ +
− + − +
 ÷
 
+ − +
+ − + − + − +
+ − +
= =
( )
( )
( )
( )
( )
2
3
2

2
x 0 x 0
2
2 2
3
3
2
x 0
2
2 2
3
3
3 4 5
x 0
3
x 0
2x 1 (x 1)
1 3x (x 1)
x
lim lim
x 2x 1 (x 1)
x 1 3x (x 1) 1 3x (x 1)
x (x 3) 1 3
lim 1
2 2
x 1 3x (x 1) 1 3x (x 1)
1 4x. 1 6x. 1 8x. 1 10x 1
*Bµi8 : lim
x
1 4x. 1 6x. 1 8x

lim
→ →
→
→
→
+ + +
+ − +
−
− =
 
+ + +
+ + + + + +
 ÷
 
− +
− = − − = −
 
+ + + + + +
 ÷
 
=
+ + + + −
+ + +
( ) ( )
( )
4 5 3 4
3 4 3 3
x 0
3 4 5 3 4
x 0 x 0

3
x 0 x 0
2
x 0
. 1 10x 1 4x. 1 6x. 1 8x
x
1 4x. 1 6x. 1 8x 1 4x. 1 6x. 1 4x. 1 6x 1 4x 1 4x 1
lim
x
1 4x. 1 6x. 1 8x. 1 10x 1 1 4x. 1 6x. 1 8x 1
lim lim
x x
1 4x 1 6x 1
1 4x 1
lim lim
x x
1 4x 1
XÐt : I lim li
x
→
→ →
→ →
→
+ − + + +
+ + + − + + + + + − + + + −
+
+ + + + − + + + −
= +
+ + −
+ −

+ +
+ −
= =
( )
x 0
n
n 5 4 3 2
x 0
4x 4
m 2
2
x 1 4x 1
1 2nx 1
Còng nh$ vËy ta cã : I lim 2 I I I I I 8
x
→
→
= =
+ +
+ −
= = ⇒ = + + + =
Page 3 of 8
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P.2512 – 34T – Hoàng Đạo Thúy Tel: (094)-2222-408
Hà Nội, ngày 28 tháng 02 năm 2010
( )
(
)
( )
(

)
( )
( ) ( )
3
2
x 0
3 3
2 2
x 0 x 0 x 0
x 0 x 0
2 2
2 2
3 3
3
4
x 7
4
4
4
t 2
2x 1 x 1
*Bµi 9 : lim
sin x
2x 1 1 x 1 1 x 1 1
2x 1 1
lim lim lim
sin x sin x sin x
2 x
lim lim 2 0 2
sin x

sin x
2x 1 1
x 1 x 1 1
x
x
x 2 x 20
*Bµi10 : lim
x 9 2
t
§ Æt t x 9 x t 9 I lim
→
→ → →
→ →
→
→
+ − +
+ − − + − + −
+ −
= = −
= − = − =
 
+ +
+ + + +
 ÷
 
+ − +
+ −
= + ⇒ = − ⇒ =
( )
(

)
( )
( )
( )
( )
(
)
( )
( )
( )
3
4
3
4 4 4
t 2 t 2 t 2
4
2
4
t 2 t 2
4
2
3
4 4
3
2
t 2
2
3
4 4
3

3
x 0
7 t 11
t 2
t 7 3 t 11 3 t 16
lim lim lim
t 2 t 2
t 2 t 7 3
t 4 t 2
t 16
lim lim
t 7 3
t 2 t 11 3 t 11 9
t 4 t 2
16 32 176
lim
3 27 27
t 11 3 t 11 9
1 4x 1 6x
*Bµi11: lim
→ → →
→ →
→
→
− − +
−
− − + − −
= − =
− −
− − +

+ +
−
− =
 
− +
− + + + +
 ÷
 
+ +
− = + =
 
+ + + +
 ÷
 
+ − +
( )
( )
( )
( )
2
2
3
2 2
2
x 0 x 0 x 0
3 2
2
x 0 x 0
2
2 2

3
3
2
x 0
2
2 2
3
3
x
1 4x (1 2x) 1 6x (1 2x) 1 4x (1 2x)
lim lim lim
x x
x 1 4x (1 2x)
1 6x (1 2x) x
lim lim
x 1 4x (1 2x)
x 1 6x (1 2x) 1 6x (1 2x)
4x (3 2x)
lim
x 1 6x (1 2x) 1 6x (1 2x)
→ → →
→ →
→
+ − + + − + + − +
= − =
+ + +
+ − + −
− =
 
+ + +

+ + + + + +
 ÷
 
− +
−
 
+ + + + + +


1 12 7
2 3 2
= − + =
÷

• BTVN NGÀY 10-06:
Page 4 of 8
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Hà Nội, ngày 28 tháng 02 năm 2010
( )
2
x 2
t 0 t 0
t 0 t 0 t 0
x 0
x 0
4 x
1 Bµi1: lim
x
cos

4
t(t 4) t(t 4)
§ Æt : t x 2 x t 2 I lim lim
t
t 1
sin
cos
4
4 2
t(t 4) t(t 4) (t 4) 16
lim lim lim
t t
sin . .
t
4 4 4
.
t
4
4
sin sinsinx
2 Bµi 2 : lim
x
sin sin sinx
lim
sin si
→
→ →
→ → →
→
→

−
−
π
+ +
= − ⇒ = + ⇒ = = −
π
π
 
+
 ÷
 
+ + +
= − = − = − = −
π π π
π
π
π
−
=
( )
( )
( ) ( )
2
x 0
2 2 2
x 0 x 0 x 0
2
2
2
2

x 0 x 0 x 0
sin sinx sinx
. . 1
nx sin x x
1 cosx cos2x
3 Bµi 3 : lim
x
cosx 1 cos 2x
1 cos x cosx cosx cos2x 1 cosx
lim lim lim
x x x
x
2 sin
cosx 1 cos 2x
1 2 cosx.sin x
2
lim lim lim
2
x
x 1 cos 2x 1 cos2x x
4.
2
→
→ → →
→ → →
 
=
 ÷
 
−

−
−
− + − −
= = +
−
= + = +
+ +
 
 ÷
 
2
1 3
1
2 2
= + =
( )
2
x 0
2
x 0
2010
2 2
x 0 x 0
2
2
n 1 2 2010
2
2
2
1 cos xcos2x cos 2010x

4 Bµi 4 : lim
x
1 cos x cosx cos x cos2x cos x cos 2x cos 2010x
lim
x
cosx. 1 cos2x
1 cos x
lim lim I
x x
nx
2 sin
1 cos nx n 1
2
XÐt I I I I I
x 2
4 nx
.
n 2
→
→
→ →
−
−
− + − + +
=
−
−
= + + +
−
= = = ⇒ = + + + =

 
 ÷
 
( )
2 2 2
1 2 3 2010
2
+ + + +
Page 5 of 8
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Hà Nội, ngày 28 tháng 02 năm 2010

( )
( ) ( )
( ) ( )
x
2
x 0 x 0
x 0 t 0 x 0
2010(2010 1)(2.2010 1)
12
ln sin x cosx
5 Bµi 5 : lim
x
ln sin x cosx ln sin x cosx
sin 2x
lim lim .
2x sin 2x 2x
ln sin x cosx ln 1 t

sin 2x
Mµ : lim lim 1 Víi t sin 2x vµ lim 1
sin 2x t 2x
I 1.1 1
→∞
→ →
→ → →
+ +
=
+
−
+ +
= =
+ +
= = = =
⇒ = =

cosx cos3x
2
x 0
cosx cos3x
2 2
x 0
cosx cos3x cos x cos3x
2 2
x 0 x 0
cosx cos3x
2 2
x 0
e cos 2x

6 Bµi 6 : lim
x
e 1 1 cos2x
lim
x x
e 1 e 1 cos x cos3x
*)Ta c ã : lim lim .
x cosx cos3x x
e 1 1 cos3x 1 cosx
lim
cos x cos3x x x
−
→
−
→
− −
→ →
−
→
−
−
 
− −
= +
 ÷
 
 
− − −
=
 ÷

−
 
− − −
 
= −

−
 
cosx cos3x t
x 0 t 0
2 2
2 2
x 0
2
x 0
x
x
x
x
e 1 e 1
. Do lim lim 1
cos x cos3x t
1 cos3x 1 cos x 3 1
lim 4
x x 2 2
1 cos2x
*)MÆt kh¸c : lim 2 I 4 2 6
x
x 3
7 Bµi 7 : lim

x 1
2 2 1
lim 1 . §Æt : x 2t 1;x t
x 1 x 1 t
−
→ →
→
→
→+∞
→+∞
− −
= =
÷
−
− −
 
− = − =
 ÷
 
−
= ⇒ = + =
+
 
−
 ÷
+
 
 
= + = ⇒ = − → +∞ ⇒ → +∞
 ÷

+ +
 
⇒
2t 1 2t 1
2
t t t
1 1 1
I lim 1 lim 1 . lim 1 e
t t t
− −
→+∞ →+∞ →+∞
     
= + = + + =
 ÷  ÷  ÷
     



Page 6 of 8
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P.2512 – 34T – Hoàng Đạo Thúy Tel: (094)-2222-408
Hà Nội, ngày 28 tháng 02 năm 2010
(
)
(
)
(
)
(
)

(
)
( )
(
)
(
)
3
3 2 2
x
3
3 2 2
x
2
3
3 2
2
x x
3
3 2 3 2 2
3
2
x
3
3
2
x x x
2
2
8 Bµi 8 : lim x 3x x x 1

lim x 3x x x x 1 x A B
3x
*)A lim x 3x x lim
x 3x x x 3x x
3
lim 1
3 3
1 1 1
x x
1
1
x 1
x
B lim x x 1 x lim lim
1 1
x x 1 x
1 1
x x
→+∞
→+∞
→+∞ →+∞
→+∞
→+∞ →+∞ →+∞
− + − − +
= + − − − + − = −
= + − =
+ + + +
= =
 
+ + + +

 ÷
 
− +
− +
= − + − = =
− + +
− + +
3
x 0
2
2
3 2
x 0 x 0 x 0
2
2
x 0
2
2 2
2
x 0 x 0
1
2
tan x sin x
9 Bµi 9 : lim
x
1
sinx x
sinx 1
(1 cos x) 2sin
1

cosx
x 2
lim lim lim
x x .cosx 2
x
4. .cos x
2
1 x cosx
10 Bµi10 : lim
x
1 x 1 cosx 1 1
lim lim
x x
1 x 1
→
→ → →
→
→ →
= −
 
 ÷
 ÷
 
−
−
 
−
−
 ÷
 

= = = =
 
 ÷
 
+ −
−
 
 
+ − −
= − =
 ÷
 ÷
 ÷
+ +
 
 
2
2
x 0
x
2sin
1 1
2
lim 1
2 2
x
4.
2
→
 

 ÷
−
 ÷
− = + =
 ÷
 
 ÷
 ÷
 
 

Page 7 of 8
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P.2512 – 34T – Hoàng Đạo Thúy Tel: (094)-2222-408
Hà Nội, ngày 28 tháng 02 năm 2010
( ) ( )
( )
3
x 0
3 3
x 0 x 0
2
2
x 0
3 2
x 1
3 2
x 1
1 tan x 1 sin x
11 Bµi11: lim

x
tan x sin x s inx(1 cos x)
lim lim
x 1 tan x 1 sin x x 1 tan x 1 sin x cosx
x
sin
sinx
2
.2
x
x
4.
1 1 1
2
lim .
2 2 4
1 tan x 1 sin x cosx
x x 2
12 Bµi12 : lim
sin(x 1)
x 1 x
lim
→
→ →
→
→
→
+ − +
−
− −

= =
+ + + + + +
 
 ÷
 
= = =
+ + +
+ −
−
−
− +
=
( )
2
2
x 1 x 1
x 1
(x 1)(x x 1) x 1 (x 1)
1 (x x 1) (x 1)
lim lim
sin(x 1)
sin(x 1) sin(x 1)
x 1
sin(x 1)
Do lim 1 I 5
x 1
→ →
→
− + + + − +
− + + + +

= =
−
− −
−
−
= ⇒ =
−
………………….Hết…………………
BT Viên môn Toán hocmai.vn
Trịnh Hào Quang
Page 8 of 8