Question

（2008•盧灣區(qū)二模）計(jì)算：

lim

n→∞

(1+

2

3n+1

)n=

e

2

3

．

Answer 1

分析：根據(jù)題意，設(shè)

3n+1

2

=t，則n=

2t-1

3

，變形可得

lim n→∞ [(1+ 1 t )t]  2 3

lim n→∞ (1+ 1 t ) 1 3

，分析可得，當(dāng)n→∞時(shí)，它的極限為e

2

3

，進(jìn)而可得答案．

解答：解：設(shè)

3n+1

2

=t，則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

故答案為：e

2

3

．

點(diǎn)評(píng)：本題考查極限的計(jì)算，需要牢記常見(jiàn)的極限的化簡(jiǎn)方法．