求((n+1)/(n+2))^3n的极限,麻烦给出重要的步骤.

求((n+1)/(n+2))^3n的极限,麻烦给出重要的步骤.
求((n+1)/(n+2))^3n的极限,麻烦给出重要的步骤.

求((n+1)/(n+2))^3n的极限,麻烦给出重要的步骤.
((n+1)/(n+2))^3n=(1-1/(n+2))^(-(n+2)*3n/(-n-2))
lim((n+1)/(n+2))^3n=lim(1-1/(n+2))^(-(n+2)*3n/(-n-2))=e^[lim(3n/(-n-2))]=e^(-3).[n->∞]

(n+1)/(n+2) = 1 - 1/(n+2)
ln[((n+1)/(n+2))^3n] = 3n ln(1- 1/(n+2)) = ln(1- 1/(n+2)) / (1/n)
lim[ 3ln(1- 1/(n+2)) / (1/n) 等价无穷小代换 ln(1- 1/(n+2)) ~ 1/(n+2)
= lim[ -3/(n+2) / (1/n)] = lim[ -3n/(n+2)] = -3
原式 = e^(-3)