怎么得来的？求详细过程

更新时间2022-02-26 21:21:17

[x→∞]lime^{[ln(1+3^n)]/n}=[x→∞]lime^{[1/(1+3^n)]*(ln3)*(3^n)/1}

当n→+∞时，ln(1+3ⁿ)→+∞，

[ln(1+3ⁿ)]/n属于0/0型，

用罗比塔法则变换。

n'=1

ln'(1+3ⁿ)=(1+3ⁿ)'/(1+3ⁿ)=(ln3)3ⁿ/(1+3ⁿ)

lim【n→+∞】e^{[ln(1+3ⁿ)]/n}

=e^lim【n→+∞】{[ln(1+3ⁿ)]/n}

=e^lim【n→+∞】[(ln3)3ⁿ/(1+3ⁿ)]

=e^(ln3)lim【n→+∞】[3ⁿ/(1+3ⁿ)]

=3