Question

（2008•卢湾区二模）计算：

lim

n→∞

(1+

2

3n+1

)n=

e

2

3

．

Answer 1

分析：根据题意，设

3n+1

2

=t，则n=

2t-1

3

，变形可得

lim n→∞ [(1+ 1 t )t]  2 3

lim n→∞ (1+ 1 t ) 1 3

，分析可得，当n→∞时，它的极限为e

2

3

，进而可得答案．

解答：解：设

3n+1

2

=t，则n=

2t-1

3

lim

n→∞

(1+

2

3n+1

)n=

lim

n→∞

(1+

1

t

)

2t-1

3

=

lim n→∞ [(1+ 1 t )t]  2 3

lim n→∞ (1+ 1 t ) 1 3

=e

2

3

故答案为：e

2

3

．

点评：本题考查极限的计算，需要牢记常见的极限的化简方法．