LIM[IN(1/X)]^X X趋于0正
1个回答

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+

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