LIM[IN(1/X)]^XX趋于0正

Question

LIM[IN(1/X)]^X X趋于0正

Answer 1

令Int(1/x)=n,当x→0+
则1/x=n+t,0＜t＜1
则[Int(1/x)]^x=n^[1/(n+t)]
因为1/(n+1)＜1/(n+t)＜1/n
所以n^[1/(n+1)]＜n^[1/(n+t)]＜n^(1/n)
所以lim{n^[1/(n+1)]}≤lim[Int(1/x)]^x≤lim[n^(1/n)],当x→0+,n→+∞
而lim[n^(1/n)]=1,n→+∞
lim{n^[1/(n+1)]},n→+∞
=e^{lim[ln(n)/(n+1)]},n→+∞
因为lim[ln(n)/(n+1)]=0,n→+∞
所以lim{n^[1/(n+1)]}=e^0=1,n→+∞
所以1≤lim[Int(1/x)]^x≤1,x→0+
即lim[Int(1/x)]^x=1,x→0+