LIM[IN(1/X)]^X X趋于0正
来源:学生作业帮 编辑:作业帮 分类:综合作业 时间:2024/10/04 19:35:39
LIM[IN(1/X)]^X X趋于0正
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+
Lim(x趋于正无穷)lnx的极限是1,
lim[(1-e^(-x))^1/2]/x x趋于0
求x趋于0时lim(e^x-1)/x
求极限,lim x趋于0 x * sin 1/x
lim(arcsinx/x)(1/x^2)(x趋于0)
lim(x趋于0)[ln(1+2x)]/x
lim趋于0((tanx-x)/(x-sinx))^(cotx-1/x)
求极限:lim[sin(1/x)+cos(1/x)]^x (x趋于正无穷)
lim(x+e的x次方)的1/x次方=什么 x趋于正无穷大
lim(x趋于0)(1+sinx)^1/x=?
lim(1/x)*limx的极限题,x趋于0
求极限 x 趋于0 lim(cosx)^1/(x^2)