lim [((x)/(x-1))-((1)/(ln x))]
x->1+
use l'hopitals rule
Subtract the fractions to get an indeterminate form:
Use L'hopital's rule to get (product rule is needed to take care of the xlnx):
This is another indeterminate form so apply L'hopital's rule again. Try taking the limit and if you get a definite answer then you're done.