作业帮求极限~lim n[e-(1+1/n)^n] n->无穷
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求极限~lim n[e-(1+1/n)^n] n->无穷
lim n[e-(1+1/n)^n] n->无穷
lim n[e-(1+1/n)^n] n->无穷
lim(n->∞) n[e-(1+1/n)^n]
=lim(n->∞) n{ e-e^[nln(1+1/n)]}
=lim(n->∞) -e*n{ e^[nln(1+1/n) - 1] - 1 }
∵(n->∞) t = [nln(1+1/n) - 1] -> 0 ,e^t -1 t
=lim(n->∞) -e* n [nln(1+1/n) - 1]
∵ ln(1+1/n) = 1/n - 1/2n^2 + o(1/n^2) ,
注：此处极限也可用罗必塔法则
=lim(n->∞) -e* [ n - 1/2 + o(1) - n ]
= e/2
=lim(n->∞) n{ e-e^[nln(1+1/n)]}
=lim(n->∞) -e*n{ e^[nln(1+1/n) - 1] - 1 }
∵(n->∞) t = [nln(1+1/n) - 1] -> 0 ,e^t -1 t
=lim(n->∞) -e* n [nln(1+1/n) - 1]
∵ ln(1+1/n) = 1/n - 1/2n^2 + o(1/n^2) ,
注：此处极限也可用罗必塔法则
=lim(n->∞) -e* [ n - 1/2 + o(1) - n ]
= e/2
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