Proving the asymptotic relationship between two functions

Question

I was playing around with numbers a few days ago and found an asymptotic approximation to two functions:

$$y=-\ln{x}$$ And $$y=x^{1-\frac{1}{x}}-x$$

Can I have a proof that it is (or isn't) asymptotic? (I'm pretty sure it is but I'm not 100% sure)

Answer 1

Hint: Write

$$ x^{1-1/x} = x e^{-\log x/x}, $$

then use the power series for $e^y$.