若limx→1x2-6x+5x2-1=a，则a=______，limn→∞(1a+1a2+1a3+…+1an)=______．

题型：石景山区一模难度：来源：

若

lim

x→1

x2-6x+5

x2-1

=a，则a=______，

lim

n→∞

(

1

a

+

1

a2

+

1

a3

+…+

1

an

)=______．

答案

∵

lim

x→1

x2-6x+5

x2-1

=

lim

x→1

(x-1)(x-5)

(x+1)(x-1)

=

lim

x→1

x-5

x+1

=-2，
∴由

lim

x→1

x2-6x+5

x2-1

=a，知a=-2．
∴

lim

n→∞

(

1

a

+

1

a2

+

1

a3

+…+

1

an

)=

lim

n→∞

1 -2 [1-(- 1 2 )n]

1- (- 1 2 )

=-

1

3

．
答案：-2，-

1

3

．