limit (infinity-infinity)

x - ln x = ln e^x - ln x = ln (e^x/x).
e^x/x = (1 + x + x^2/2! + x^3/3! + .....)/x = 1/x + 1 + x/2! + x^2/3! + .....
Based on the above
lim (x - ln x) = lim (ln e^x - ln x) = lim [ln (e^x/x)]
= ln lim (1/x + 1 + x/2! + x^2/3! + ....)
so when x tends to infinity, 1/x + 1 + x/2! + x^2/3! + .... tends to infinity,
so ln lim ( 1/x + 1 + x/2! + ....) = lim (x - ln x) tends to infinity.