求一道极限题lim(sinn!×n^2/3）/(n+1)

Question

求一道极限题
lim(sinn!×n^2/3）/(n+1)

Answer 1

解法一：(定义法)
∵对任意的ε>0,存在N=[1/ε³]（[1/ε³]表示不超过1/ε³的最大整数）,当n>N时,
有|n^(2/3)sinn!/(n+1)|≤n^(2/3)/(n+1)＜n^(2/3)/n=n^(-1/3)＜ε
∴根据极限定义,知lim(n->∞)[n^(2/3)sinn!/(n+1)]=0；
解法二：（两边夹法）
∵|n^(2/3)sinn!/(n+1)|≤n^(2/3)/(n+1)
∴-n^(2/3)/(n+1)≤n^(2/3)sinn!/(n+1)≤n^(2/3)/(n+1)
∵lim(n->∞)[n^(2/3)/(n+1)]=lim(n->∞)[(1/n^(1/3))/(1+1/n)]=0
同理lim(n->∞)[-n^(2/3)/(n+1)]=0
∴根据两边夹定理,知lim(n->∞)[n^(2/3)sinn!/(n+1)]=0.