lim((1+x)^(1/x)/e))^(1/x)x趨近於0

lim((1+x)^(1/x)/e))^(1/x)x趨近於0


lim(x->0)[(1+x)^(1/x)/ e ]^(1/x)
=lim(x->0)[(1+x)/ e^x ]^(1/x^2)
let
L = lim(x->0)[(1+x)/ e^x ]^(1/x^2)
lnL = lim(x->0)ln[(1+x)/ e^x ] / x^2
=lim(x->0)[ ln(1+x)- x ] / x^2(0/0)
=lim(x->0)[ 1/(1+x)-1]/(2x)
=lim(x->0)-1/[2(1+x)]
= -1/2
L = e^(-1/2)