lim³√n^sinn!÷(n+1)

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卡夫饿不

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摘要亲亲，您好，很高兴为您解答您咨询的问题已为您找到：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.

咨询记录 · 回答于2022-09-25

lim³√n^sinn!÷(n+1)

亲亲，您好，很高兴为您解答您咨询的问题已为您找到：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.

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