Find the limit of function (x / x-1) - (1 / LNX) x tending to 1

Find the limit of function (x / x-1) - (1 / LNX) x tending to 1

t=x-1
T=(t+1)/t-1/ln(t+1),t->0
ln(1+t)=t-t^2/2.
T = 1 + 1 / T-1 / [T-T ^ 2 / 2 + 0 (T ^ 3)] --- general score
=1+[1-t/2+0(t^2)-1]/[t-t^2/2+0(t^3)]
=1+[t/2+0(t^2)]/[t-t^2/2+0(t^3)]
=1+1/2
=3/2