BT gioi han ham so
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Bµi tËp chuyªn ®Ò: Giíi h¹n hµm sè Duy Chung - THPT Mª Linh
gioi han ham so-Bai tap-2011.doc - 1 - 12/14/2011
I Giíi h¹n d¹ng
0
0
1.
2
2
3
5 6
lim
8 15
x
x x
x x
2.
3 2
4 2
2
2 4 8
lim
8 16
x
x x x
x x
3.
0
(1 )(1 2 )(1 3 ) 1
lim
x
x x x
x
4.
3
4 2
1
3 2
lim
4 3
x
x x
x x
5.
5
2 5
0
(1 ) (1 5 )
lim
x
x x
x x
6.
3
2
4 2
lim
2
x
x
x
7.
3 3
0
1 1
lim
x
x x
x
8.
3
1
1
lim
1
x
x
x
9.
2
2
1
3 2 4 2
lim
3 2
x
x x x
x x
10.
4
3
1
1
lim
1
x
x
x
11.
3
8
2 4
lim
2
x
x
x
12.
1
2 2 1 3
lim
1
x
x x
x
13.
2 2
2
1
2 6 2 2
lim
4 3
x
x x x x
x x
14.
2
2
0
1 1
lim
16 4
x
x
x
15.
1
2 1
lim
1
x
x x
x
16.
3
1
3 2
lim
1
x
x x
x
17.
2
3
1
3 2
lim
1
x
x
x
18.
2
3
2
4
lim
2 3 2
x
x
x
19.
3 2
3
2
1
2 1
lim
1
x
x x x
x
20.
0
9 16 7
lim
x
x x
x
21.
4
7
9 2
lim
7
x
x
x
22.
3
2
1
7 3
lim
3 2
x
x x
x x
23.
0
2 1 8
lim
x
x x
x
24.
3
0
2 1 1 3
lim
x
x x
x
25.
3
0
1 1
lim
x
x x
x
26.
3
2
2
11 8 43
lim
2 3 2
x
x x
x x
27.
33 2
2
1
5 7
lim
1
x
x x
x
28.
3
2
2
8 11 7
lim
3 2
x
x x
x x
29.
3 2
4
2
0
1 1 2
lim
x
x x
x x
30.
3
3 2
1
7 3
lim
1
x
x x
x
31.
5
4
1
2 1 2
lim
1
x
x x
x
32.
43
0
1 1
3 4
lim
1 1
2
x
x x
x
33.
34 6
1
4 5 8 21
lim
1
x
x x
x
34.
3 5
4
0
1 4 . 1 6 . 1 8 . 1 10 1
lim
x
x x x x
x
35.
2
3
0
1 1 1
lim
x
x x
x
36.
3 2
0
2 1 1
lim
sin
x
x x
x
37.
3
4
7
2 20
lim
9 2
x
x x
x
38.
3
2
0
4 1 1 6
lim
x
x x
x
39.
3 3 44
0
27 1 81 1
lim
x
x x
x
Bµi tËp chuyªn ®Ò: Giíi h¹n hµm sè Duy Chung - THPT Mª Linh
gioi han ham so-Bai tap-2011.doc - 2 - 12/14/2011
II Giíi h¹n d¹ng
40.
3 2
4 3 2
2 3 4 1
lim
5 2 3
x
x x x
x x x
41.
5 3
5 4 2
5 7 4 3
lim
5 7 1
x
x x x
x x x
42.
2 2
2
3 (2 1)(3 2)
lim
2 1 4
x
x x x x
x x
43.
32 2
54 4
4
9 9 6 5
lim
16 3 8 7
x
x x
x x
44.
2 2
3 2
(2 3) .(4 7)
lim
(3 1).(10 9)
x
x x
x x
45.
30 25
55
(2 3) .( 7)
lim
(3 1)
x
x x
x
46.
4
( 1)( 2)( 3)( 4)
lim
(3 1)
x
x x x x
x
47.
2
3 3 2
4 3 7
lim
27 5 4
x
x x
x x x
48.
5 3 1
lim
1
x
x x
x
49.
2
2
lim
3 1
x
x x x
x x
50.
35 24
5 4 3
1 2
lim
1 2
x
x x
x x
51.
2
2
4 5 2 1
lim
3 2 7
x
x x x
x x x
III Giíi h¹n d¹ng
52. lim
x
x x x x
53. lim
x
x x x x
54. lim
x
x x x x x x
55.
2
lim 2 5 4 4 1
x
x x x
56.
3 3
3 2 3
lim 7 8
x
x x x x
57.
3
4
lim ( 5)( 6)( 7) ( 1)( 2)( 3)( 4)
x
x x x x x x x
58.
2 2
lim 1 1
x
x x x x
59.
2
2
33
lim 1 ( 1)
x
x x
60.
3
2
lim 1
x
x x
61.
2 2
lim 4 7 1 4 8 1
x
x x x x
62.
3 2
lim 2 1
x
x x
IV Giíi h¹n d¹ng
0.
63.
2
lim . 1
x
x x x
64.
32 4 6
lim . 9 7 27 5
x
x x x
65.
3
3 3
2
lim . 1 1
x
x x x
66.
32 9 124
lim . 8 1 16 3
x
x x x
67.
2 2
lim . 2 2
x
x x x x x x
68.
2
lim . 2
x
x x x
69.
2
lim . 4 9 2
x
x x x
70.
2
lim . 9 1 2
x
x x x
71.
32 3
lim . 1 1
x
x x x
72.
2 4 4
lim . 3 5 2
x
x x x
73.
32 2 3
lim . 1 5 2
x
x x x x
Bµi tËp chuyªn ®Ò: Giíi h¹n hµm sè Duy Chung - THPT Mª Linh
gioi han ham so-Bai tap-2011.doc - 3 - 12/14/2011
V Giíi h¹n lîng gi¸c: cho
0
sin
lim 1
x
x
x
1.
1
lim ( 1) sin
1
x
x
x
2.
2
1
1
lim ( 1) sin
1
x
x
x
3.
2
0
1 cos
lim
x
x
x
4.
0
1 cos
lim
x
x
x
5.
2
0
1 cos 5
lim
x
x
x
6.
2
0
1 cos
lim
2 tan
x
x
x
7.
3
0
2sin sin 2
lim
x
x x
x
8.
2
0
1 cos
lim
sin
x
x
x
9.
0
1 cos 4
lim
1 cos 5
x
x
x
10.
3
0
tan sin
lim
x
x x
x
11.
0
1 sin cos
lim
1 sin cos
x
ax ax
bx bx
12.
3
0
tan sin
lim
x
ax ax
x
13.
3
sin 3 cos
lim
sin 3
x
x x
x
14.
4
sin cos
lim
sin
4
x
x x
x
15.
2
0
1 cos cos 2
lim
x
x x
x
16.
2
0
cos cos
lim
sin
m n
x
ax bx
cx
17.
lim 2 2 2 2 2
n
n
n dÊu c¨n
18.
4
2
sin
2
lim
tan 1
x
x
x
19.
4
lim 4 tan 2
x
x x
20.
6
sin
6
lim
3
cos
2
x
x
x
21.
2
2
4
lim
cos
4
x
x
x
22.
1
lim(1 ) tan
2
x
x
x
23.
3
sin 3
lim
1 2cos
x
x
x
24.
3
2
0
3 sin
lim
1 cos
x
x x x
x x
25.
3
0
1 cos
lim
.sin 2
x
x
x x
26.
2
3 3
0
1 cos
lim
tan sin
x
x
x x
27.
0
sin sin
lim
sin
x
ax bx
cx
28.
2
0
cos cos
lim
sin
x
ax bx
cx
29.
0
1 cos
lim
1 cos
x
ax
bx
30.
0
cos 7 cos3
lim
cos15 cos11
x
x x
x x
31.
2
2
0
(1 cos )
lim
sin 2 .tan
x
x x
x x
32.
0
1 cos 4
lim
.sin
x
x
x x
33.
2
0
2 1 cos
lim
sin
x
x
x
34.
0
1 sin 1 sin
lim
tan
x
x x
x
35.
2
0
lim
1 sin cos
x
x
x x x
36.
0
1 3 1
lim
sin sin 3
x
x x x
Bµi tËp chuyªn ®Ò: Giíi h¹n hµm sè Duy Chung - THPT Mª Linh
gioi han ham so-Bai tap-2011.doc - 4 - 12/14/2011
37.
2
0
sin( 2 ) 2cot( ) cot
lim
x
a x a x a
x
38.
0
3 2sin
lim
1 cos
x
x x
x
39.
0
2
1 sin cos 2
lim
tan
2
x
x x x
x
40.
0
9 3
lim
sin 7
x
x
x
41.
2
0
1 sin cos 2
lim
sin
x
x x x
x
42.
2
0
sin
lim
1 sin cos
x
x
x x x
43.
0
sin 5
lim
3 3
x
x
x
44.
2
2
0
2sin sin 1
lim
2sin 3sin 1
x
x x
x x
45.
0
1 cos .cos 2 .cos3
lim
1 cos
x
x x x
x
46.
3
3
tan 3tan
lim
cos
6
x
x x
x
47.
3
3
4
1 cot
lim
2 cot cot
x
x
x x
48.
3
2
0
1 cos cos 2 . cos 3
lim
x
x x x
x
49.
0
1 |1 sin 3 |
lim
1 cos
x
x
x
50.
0
2
lim cot
sin 2
x
x
x
51.
0
2
cos cos
2
lim
sin
2
x
x
x
52.
3
0
tan sin
lim
x
x x
x
53.
0
1 sin 2 cos 2
lim
1 sin 2 cos 2
x
x x
x x
54.
2
2
0
tan( ) tan( ) tan
lim
x
a x a x a
x
55.
3
0
1 tan 1 sin
lim
x
x x
x
56.
0
1 2 1 sin
lim
3 4 2
x
x x
x x
57.
3 2
2
1
2
lim
1 sin( 1)
x
x x
x x
58.
2
2
0
1 cos
lim
x
x x
x
59.
0
1 cos
lim
1 cos
x
x
x
60.
2
0
2006 1 cos 3 .cos 5 .cos 7
lim .
2005 sin 7
x
x x x
x
V Giíi h¹n d¹ng:
1
1.
4 3
1
lim
2
x
x
x
x
2.
2
cot
2
0
lim 1
x
x
x
3.
1
sin
0
1 tan
lim
1 sin
x
x
x
x
4.
1 1
lim sin cos
x
x
x x
5.
2 1
2
lim
1
x
x
x
x
6.
2
2
2
3
lim
2
x
x
x
x
7.
1
lim 1 sin
x
x
8.
cot
0
1 sin cos
lim
1 sin cos
x
x
x ax
x bx
