LIM[IN(1/X)]^X X趋于0正
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LIM[IN(1/X)]^X X趋于0正
![LIM[IN(1/X)]^X X趋于0正](/uploads/image/z/20189446-70-6.jpg?t=LIM%5BIN%281%2FX%29%5D%5EX+X%E8%B6%8B%E4%BA%8E0%E6%AD%A3)
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+
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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