当x趋向于正无穷，求lim（1+(2^n)+(3^n)）^(1/n)的极限

Question

如题所述

Answer 1

以前做过
提供2种解法
解1：n->无穷
3^n<(1+2^n+3^n)<3*3^n
lim
(3^n)^(1/n)=3且lim
(3*3^n)^(1/n)=3
由夹逼准则知lim(1+2^n+3^n)^(1/n)=3
解2
n→∞
lim(1^n+2^n+3^n)^(1/n)
=e^lim[(1/n)*ln(1^n+2^n+3^n)]
下面求lim[(1/n)*ln(1^n+2^n+3^n)]
=lim(1/n)*ln{(3^n)*[(1/3)^n+(2/3)^n+1]}
=lim(1/n)*{nln3+ln[1+(1/3)^n+(2/3)^n]}
这里ln[1+(1/3)^n+(2/3)^n]等价于(1/3)^n+(2/3)^n
=ln3+im[(1/3^n+(2/3)^n]/n
=ln3
所以最后结果为e^ln3=3