Why are these two functions so close in value? [closed]
I don't think there is anything special about the fact that the function $f(x) = ln(1 + 1/x)$ is -for large $x$- so well approximated by your expression. $f(x)$ is well-behaved and monotically decreassing. Its Taylor series for large $x$ can be derived. It is then possible to construct other functions $g(x)$ that have the same first few terms of the Taylor series. For example, I expect this function to work even better:
$$g(x) = \frac{x+1/6}{x(x+2/3)}$$
The reason is that the error between $f$ and $g$ for large $x$ is now of order $x^{-4}$.