lim(x->1)[ x/(x-1) - 1/lnx ]
=lim(x->1) [xlnx-(x-1)]/[(x-1)lnx] (0/0)
= lim(x->1) [ (1+lnx-1) / (lnx + (x-1)/x) ]
= lim(x->1) [ xlnx/(xlnx+(x-1) ] (0/0)
= lim(x->1) [ (lnx+1)/(x+1+1) ]
=1/2