How might one prove two functions asymptotic?

Question

Lets say that function $f$ and $g$ look to be converging on one another. What tests might you use to prove that they are/are not asymptotic?

If you want an example to work on, lets say $f(x)=\ln x$ and $g(x)=x-x^{1-1/x}$

I know that these functions are asymptotic, but have no way of proving it. Thanks to all help in advance.

Answer 1

As $x \to \infty$,

$\begin{align*} x^{1-1/x} &=e^{\ln x(1-1/x)}\\ &=e^{\ln x}e^{-\ln x/x}\\ &=x(1-\ln x/x + (\ln x)^2/(2 x^2)+o(1/x^2))\\ &=x-\ln x+ (\ln x)^2/(2 x)+o(1/x)\\ \end{align*} $

so

$\begin{align*} g(x) &=x-x^{1-1/x}\\ &=x-(x-\ln x+ (\ln x)^2/(2 x)+o(1/x))\\ &=\ln x- (\ln x)^2/(2 x)+o(1/x)\\ &\approx \ln x\\ \end{align*} $

I believe that this is what you want.