高数简单求极限lim[(3√n^2)*sin ]/(n+1) n--∞n的3/2次方乘以sin( n的阶乘) 除以 n+
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高数简单求极限
lim[(3√n^2)*sin ]/(n+1) n--∞
n的3/2次方乘以sin( n的阶乘) 除以 n+1 n趋于无穷
lim[(3√n^2)*sin ]/(n+1) n--∞
n的3/2次方乘以sin( n的阶乘) 除以 n+1 n趋于无穷
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.
∵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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