KIỂM TRA 45’
MÔN: GIẢI TÍCH 11 (cơ bản)
Câu 1:(6.0 điểm) Tính các giới hạn sau:
( )
) lim 2 2
n
a n n
→+∞
+ − −
;
2
2
7 10
)lim
2
x
x x
b
x
→
− +
−
;
3
2
1
2
)lim
1
x
x x

2
3 2
2
( )
2
2
x x
khi x
f x
x
x b khi x

− +
>

=
−


+ ≤


a) Tìm
( ) ( )
2 2
lim ; lim ;
x x
f x f x
− +
→ →

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ĐÁP ÁN
Câu Nội dung Điểm
1a
( )
2 2
lim 2 2 lim
2 2
n n
n n
n n
n n
→+∞ →+∞
+ − +
+ − − =
+ + −
4
4 0
lim lim 0
2
2 2 2 2
1 1
n n
n
n n
n n
→+∞ →+∞
= = = =

1c
( )
( )
( ) ( )
2
3
2
1 1
1 2
2
lim lim
1 1 1
x x
x x x
x x
x x x
→ →
− + +
+ −
=
− + −
( )
( )
2
1
2
1 1 2
lim 2
1 1 1
x

( )
( )
( )
( )
( )
2 2
2 2
2 2 2 1
lim lim
24 12
2 2 4 1 3 2 4 1 3 2 2 4.2 1 3
x x
x
x x x x x
→ →
−
= = = = =
− + + + + + + + + +
0.75
0.75
2
Các số trong dãy tổng lập thành một CSN lùi vô hạn với công bội
1
2
q = −
và
1
1
2
u = −

→ →
= + = +
( ) ( )
( )
2
2 2 2 2
1 2
3 2
lim ( ) lim lim lim 1 1
2 2
x x x x
x x
x x
f x x
x x
+ + + +
→ → → →
− −
− +
= = = − =
− −
0.5
1.0
3b Hàm số đã cho liên tục tại mọi x khác 2
Hàm số đã cho liên tục trên
¡
khi và chỉ khi hàm số liên tục tại x = 2
2 2
lim ( ) lim ( )
x x