Inverse of approximations

Question

Suppose over some $A\subseteq \mathbb{R}$ we have a function, $f:A\to\mathbb{R}$. We also have a function, $g:\mathbb{R}\to\mathbb{R}$, that approximates $f$, by which we mean that $$ \lim_{x\to\infty} \frac{f(x)}{g(x)}=1$$ Now suppose that both $f$ and $g$ are invertible. Given an expression for $g^{-1}$, can we say anything about $f^{-1}$? Intuitively, if two functions approximate each other, it seems that their inverses ought to as well, however this is not the case as for $f(x)=\ln(x)$ and $g(x)=\ln(x)+1$ we find that $f \approx g$, but $$\lim_{x\to\infty} \frac{f^{-1}(x)}{g^{-1}(x)}=e.$$ So then, can anything be said about the inverses of two approximating functions?