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Bài tập về giới hạn 01

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ξ2. GIỚI HẠN HÀM SỐ
1. Dùng đònh nghóa, CMR:
a)
x 2
lim(2x 3) 7

+ =
b)
x 3
x 1
lim 1
2(x 1)

+
=

c)
2
x 1
x 3x 2
lim 1
x 1

− +
= −

2. Tìm các giới hạn sau
a)
3 2
x 0
lim(x 5x 10x)



+ +
b)
2
x 1
x 5x 6
lim
x 2

− +

c)
x 3
lim x 1


d)
2
2
x 2
2x 3x 1
lim
x 4x 2
→−
+ +
− + +
e)
3
x 1
1 1

lim
1 x
1 2x

 

 ÷
+
 − 
f)
2
3
x 0
x 4
lim
x 3x 2


− +
g)
x 1
1 x 1 x
lim
x

+ − −
h)
x
2
sin x

lim
x
π

i)
0
1
lim
cos
x
x

j)
0
tan sin2x
lim
cos
x
x
x

+
k)
x
4
tgx
lim
x
π


π −
 Dạng vô đònh
0
0
3. Tìm các giới hạn sau:
a)
2
2
x 2
x 4
lim
x 3x 2


− +
b)
2
2
x 1
x 1
lim
x 3x 2
→ −

+ +
c)
2
2
x 5
x 5x

lim
x 25



d)
2
2
x 2
x 2x
lim
2x 6x 4


− + −
e)
3
4
x 1
x 3x 2
lim
x 4x 3

− +
− +
f)
3 2
2
x 1
x x x 1

lim
x 3x 2

− − +
− + −
g)
2
3
2
2 6
lim
8
x
x x
x
→ −
+ −
+
h)
4 2
2
3
72
lim
2 3
x
x x
x x

− −

− −
i)
5
3
1
1
lim
1
x
x
x
→−
+
+
j)
3 2
4 2
x 3
x 5x 3x 9
lim
x 8x 9

− + +
− −
k)
4 3 2
3 2
x 1
2x 8x 7x 4x 4
lim

3x 14x 20x 8

+ + − −
+ + +
l)
3 2
3
x 2
x 3x 9x 2
lim
x x 6

→ −
− − +
− +
m)
2
1
2 1
lim
1 1
x
x x

 

 ÷
− −
 
n)

3
1
1 3
lim
1 1
x
x x

 

 ÷
− −
 
o)
5 6
2
x 1
x 5x 4x
lim
(1 x)

− +

p)
3 3
h 0
(x h) x
lim
h


+ −
q)
2
3 3
x a
x (a 1)x a
lim
x a

− + +

r)
4 4
x a
x a
lim
x a




s)
3 3
h 0
2(x h) 2x
lim
h

+ −
t)

2 2
x 1
x 2 x 4
lim
x 5x 4 3(x 3x 2)

 
+ −
+
 ÷
− + − +
 
u)
1992
1990
x 1
x x 2
lim
x x 2

+ −
+ −
k)
n
2
x 1
x nx n 1
lim
(x 1)


− + −

4. Tìm các giới hạn sau:
A =
8x
18xx4
lim
3
2
2x

−+

B =
2
2
x 5
x x 30
lim
2x 9x 5

+ −
− −
C =
3 2
x 1
x 1
lim
x 2x x 2
→−

+
+ − −
D =
2
3 2
1
x
2
4x 1
lim
4x 2x 1


+ −

E =
2
2
x 1
x 4x 3
lim
x 2x 3

− +
+ −
F =
2
2
1
x

2
2x 5x 2
lim
4x 1

− +

G =
2
2
x 1
2x 3x 1
lim
x 4x 5
→−
+ +
− + +
H =
4
2
x 2
x 16
lim
x 2x
→−

+

I =
3

2
x 1
x 1
lim
x x



J =
3x4x
27x
lim
2
3
3x
+−


K =
3 2
2
x 2
x 6x 12x 8
lim
x 4x 4

− + − +
− +
L =
3 2

2
x 1
x x x 1
lim
x 5x 6

− + −
− − +
M =
3
2
x 2
8x 64
lim
x 5x 6


− +
N =
3 2
3
x 2
x 2x 6x 4
lim
8 x

+ − −

O =
3 2

2
x 2
x x 5x 2
lim
x 3x 2

+ − −
− +
P =
3 2
2
x 1
x 4x 6x 3
lim
x x 2
→−
+ + +
− −
Q =
3
2
x 1
x 3x 2
lim
x 2x 1

− +
− +
R =
5

3
x 1
x 1
lim
x 1



5. Tìm caùc giôùi haïn sau:
a)
2
x 0
x 1 x x 1
lim
x

+ − + +
b)
2
x 7
x 3 2
lim
49 x

− −

c)
2
x 2
2 x 2

lim
x 3x 2

− +
− +
d) EMBED
Equation.DSMT4
2
x 2
4x 1 3
lim
x 4

+ −

e) EMBED Equation.DSMT4
3 2
x 1
2x 7 3
lim
x 4x 3

+ −
− +
f) EMBED Equation.DSMT4
x 4
x 5 2x 1
lim
x 4


+ − +


g) EMBED Equation.DSMT4
2
2
1
2 3
lim
3 2
x
x
x x

− +
− + −
h) EMBED Equation.DSMT4
3
2
2
lim
8
x
x x
x

− +

i)
2

2
x 1
3x 2 4x x 2
lim
x 3x 2

− − − −
− +
j) EMBED Equation.DSMT4
x 4
3 5 x
lim
1 5 x

− +
− −
k) EMBED Equation.DSMT4
x 1
3 8 x
lim
2x 5 x

− +
− −
l) EMBED Equation.DSMT4
x 2
x x 2
lim
4x 1 3


− +
+ −

EMBED Equation.DSMT4
2
3
1
2 6 4 1
) lim
2 1
x
x x x
m
x x

+ + − +
− +
n) EMBED Equation.DSMT4
4
3 2
x 1
x 1
lim
x x 2


+ −

o) EMBED Equation.DSMT4
3

2
0
1 1
lim
2
x
x
x x

− −
+
p) EMBED Equation.DSMT4
3
2
1
1
lim
2 5 3
x
x
x x
→−
+
+ +
q) EMBED Equation.DSMT4
3
2
x 2
2x 12 x
lim

x 2x
→−
+ +
+
r) EMBED Equation.DSMT4
3
x 1
x 7 2
lim
x 1

+ −

s)
EMBED Equation.DSMT4
3
0
1 1
lim
1 1
x
x
x

+ −
+ −
t) EMBED Equation.DSMT4
3
x 1
x 7 2

lim
x 1

+ −

v) EMBED Equation.DSMT4
3
4
x 1
x 1
lim
x 1



w) EMBED Equation.DSMT4
3
3
x 1
x 1
lim
4x 4 2


+ −

x) EMBED Equation.DSMT4
3
2
3

2
x 1
x 2 x 1
lim
(x 1)

− +

6. Tính caùc giôùi haïn sau:
a.
x 0
x 1 x 4 3
lim
x

+ + + −
b.
x 0
x 9 x 16 7
lim
x

+ + + −
c.
3
x 0
x 1 x 4 3
lim
x


+ + + −
d.
3
x 0
x 1 x 1
lim
x

+ − +
e.
3
2
1
3 3 5
lim
1
x
x x
x

+ − +

f.
3
2
x 1
8x 11 x 7
lim
x 3x 2


+ − +
− +
 Daïng voâ ñònh


7.Tìm caùc giôùi haïn sau:
a)
x
2x 1
lim
x 1
→+∞
+

b)
2
2
x
x 1
lim
1 3x 5x
→−∞
+
− −
c)
2
x
x x 1
lim
x x 1

→+∞
+
+ +
d)
2
2
x
3x(2x 1)
lim
(5x 1)(x 2x)
→−∞

− +
e)
3
3 2
3 2 2
lim
2 2 1
x
x x
x x
→±∞
− +
− + −
f)
3 2
4
3 2 1
lim

4 3 2
x
x x
x x
→±∞
− −
+ −
g)
3 2
2
2 2
lim
3 1
x
x x
x x
→±∞
− −
− −
h)
4 2
3
3 1
lim
2 2
x
x x
x x
→±∞
− +

− + −
i)
2 2
4
x
(x 1) (7x 2)
lim
(2x 1)
→±∞
− +
+
j)
2 3
2 2
x
(2x 3) (4x 7)
lim
(3x 4) (5x 1)
→±∞
− +
− −
k)
2
x
4x 1
lim
3x 1
→∞
+


l)
2
3 2
lim
3 1
x
x x x
x
→+∞
− +

m)
2
3 2
lim
3 1
x
x x x
x
→−∞
− +

n)
2
2
x
x x 2 3x 1
lim
4x 1 1 x
→±∞

+ + + +
+ + −
o)
2
2
x
4x 2x 1 2 x
lim
9x 3x 2x
→±∞
− + + −
− +
p)
2
2
x
x 2x 3 4x 1
lim
4x 1 2 x
→±∞
+ + + +
+ + −
q)
2
x
x x 3
lim
x 1
→+∞
+

+
r)
3
3 2
2
lim
2 2
x
x x x
x
→−∞
+ +

s)
33 2 2 3 2 2
3
2
( 2 ) 2
lim
3 2
x
x x x x x x
x x
→−∞
+ + + +

t)
x
(x x x 1)( x 1)
lim

(x 2)(x 1)
→+∞
+ − +
+ −
 Daïng voâ ñònh
∞ −∞
8.Tính caùc giôùi haïn sau:
a)
)32(lim
3
xx
x

+∞→
b)
3
lim (2 3 )
x
x x
→±∞

c)
2
lim 3 4
x
x x
→±∞
− +
d)
2

x
lim ( x x x)
→−∞
+ −
e)
2
x
lim ( x x x)
→+∞
+ −
f)
)23(lim
2
xxx
x
−+−
+∞→
g)
)23(lim
2
xxx
x
−+−
−∞→
h)
2
lim ( 2 4 )
x
x x x
→±∞

− + −
i)
)22(lim −−+
+∞→
xx
x
j)
2 2
x
lim ( x 4x 3 x 3x 2)
→±∞
− + − − +
k)
2
lim ( 5 )
x
x x x
→±∞
+ +
l)
2
x
lim (2x 1 4x 4x 3)
→±∞
− − − −
m)
2
x
lim (3x 2 9x 12x 3)
→±∞

+ − + −
n)
)223(lim
2
−++−
+∞→
xxx
x

o)
)223(lim
2
−++−
−∞→
xxx
x
p)
2
lim ( 3 2 1)
x
x x x
→±∞
− + + −
q)
2
lim ( 3 1 3)
x
x x x
→±∞
− + − +

r)
2
lim ( 4 3 2 1)
x
x x x
→±∞
− + − +
s)
3
3 2
x
lim ( x x x)
→±∞
+ −
t)
3
3 2
x
lim ( x x x x)
→±∞
− + +
v)
3
2 3
x
lim ( x 1 x 1)
→+∞
+ − −
w)
3 3 2

lim ( 2 1 3 )
x
x x x x
→±∞
+ − − −
 Giới hạn một bên
9. Tìm các giới hạn sau
a)
2
2
2
lim
3 1
x
x x
x



+
b)
2
3 1
lim
2
x
x
+



c)
1
1
lim
1
x
x
x
+



d)
1
1
lim
1
x
x
x




e)
2 3
x 0
x x
lim
2x

+

+
f)
2 3
x 0
2x
lim
4x x
±

+
g)
2
33
lim
2
2

+−


x
xx
x
h)
2
33
lim
2

2

+−
+

x
xx
x
i)
4
3
lim
4
x
x
x
±



j)
2
33
lim
2
2
2
−+
+−


−→
xx
xx
x
k)
2
33
lim
2
2
2
−+
+−
+
−→
xx
xx
x
l)
3
2
x 1
x 3x 2
lim
x 5x 4


− +
− +
g)

x 0
1 x
lim x
x
±

 

 ÷
 ÷
 
h)
2
x 1
x x 2
lim
x 1
+

+ −

i)
x
2
1 cos2x
lim
x
2
+
π


+
π

10. Tìm giới hạn bên phải, giới hạn bên trái của hs f(x) tại x
o
và xét xem hàm số có giới hạn tại x
o
không ?

2
2
o
x 3x 2
(x 1)
x 1
a) f(x)
x
(x 1)
2
với x 1

− +
>



=



− <


=

2
o
4 x
(x 2)
b) f(x)
x 2
1 2x (x 2)
với x 2


<

=
 −

− >

=

3
1 x 1
x 0
c) f (x)
1 x 1
3 / 2 x 0

0
o



với x

+ −
>

=

+ −



=

11. Tìm A để hàm số sau có giới hạn tại x
o
:
a)
3
x 1
(x 1)
f(x)
x 1
Ax 2 (x 1)



<

=
 −

+ ≤

với x
0
= 1 b)
3 2
2
x 6 2x 9
A x 3
f (x)
x 4x 3x
3x 2 x 3



+ + −
+ <

=
− +


− ≥

với x

0
= 3
 Giới hạn hàm lượng giác
12. Tính các giới hạn sau:
a)
x 0
sin5x
lim
3x

b)
2
x 0
1 cos2x
lim
x


c)
2
x 0
cosx cos 7x
lim
x


d)
2
x 0
cosx cos3x

lim
sin x


e)
3
x 0
tgx sin x
lim
x


f)
x 0
1 3
lim x
sin x sin3x

 

 ÷
 
g)
0
sin2 sin
lim
3sin
x
x x
x


+
h)
0
1 sin cos2
lim
sin
x
x x
x

− −

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