limx→0(x+e^x)^(1/x)详细步骤!
问题描述：
limx→0(x+e^x)^(1/x)详细步骤!
答
原式=lim(x->0) [e^x*(1+xe^(-x))]^(1/x)
=lim(x->0) e*[1+xe^(-x)]^(1/x)
=lim(x->0) e*{[1+xe^(-x)]^(e^x/x)}^[e^(-x)]
=e*e^1
=e^2
答
设y=(x+e^x)^(1/x)则：y^x=x+e^xxlny=ln(x+e^x)lny=[ln(x+e^x)]/xlim(x->0)lny=lim(x->0)=lim(x->0)[ln(x+e^x)]/x=lim(x->0) (1+e^x)/(x+e^x)=2lim(x->0)y=e^2即：lim(x->0) (x+e^x)^(1/x)=e^2