limx趋于1(x/x-1)-(1/lnx)

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limx趋于1(x/x-1)-(1/lnx)

limx趋于1(x/x-1)-(1/lnx)
limx趋于1(x/x-1)-(1/lnx)

limx趋于1(x/x-1)-(1/lnx)
当x-->1时,lnx=ln(1+x-1)与x-1等价,于是
lim(x-->1)(x/x-1)-(1/lnx)
=lim(x-->1)[xlnx-x+1]/[(x-1)lnx]
=lim(x-->1)[xlnx-x+1]/[(x-1)^2] (利用罗比达法则)
=lim(x-->1)(lnx)]/[2(x-1)] (再用替换)
=lim(x-1)]/[2(x-1)]=1/2.

lim_{x->1}[x/(x-1) - 1/ln(x)]
=lim_{x->1}{[xln(x)-x+1]/[(x-1)ln(x)]}
=lim_{x->1}{[1+ln(x)-1]/[(x-1)/x + ln(x)]}
=lim_{x->1}{xln(x)]/[(x-1)+xln(x)]}
=lim_{x->1}{[1+ln(x)]/[1+ln(x)+1]}
=1/2