Question

若

lim

x→1

x2-6x+5

x2-1

=a，則

lim

n→∞

(

1

a

+

1

a2

+

1

a3

+…+

1

an

)的值為（　�。�

• A、-2
• B、-

  1

  3

• C、-

  1

  2

• D、3

Answer 1

分析：先由題設(shè)條件求出a的值，再求出

lim

n→∞

(

1

a

+

1

a2

+

1

a3

+…+

1

an

)的極限值．

解答：解：∵a=

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．
∴

lim

n→∞

(

1

a

+

1

a2

+

1

a3

+…+

1

an

)=

lim

n→∞

[(-

1

2

)+(-

1

2

)2+(-

1

2

)3+…+(-

1

2

)n]
=

lim

n→∞

(- 1 2 )× [1-(- 1 2 )n]

1+ 1 2

=-

1

3

．
故選B．

點(diǎn)評(píng)：本題考查極限的求法，解題時(shí)要認(rèn)真審題，仔細(xì)求解，注意公式的合理運(yùn)用．