Consider a composite function $f(g(x))$ for $x\in\mathbb{R}$. We want to approach this function for $x\rightarrow0$, and we suppose that near $0$ we have $g(x)=\mathcal{O}(g_0(x))$ for some function $g_0$. In fact: $$\tag{1}\frac{g(x)}{g_0(x)}=1+\alpha x=1+\mathcal{O}(x)$$ for some $\alpha\in\mathbb{R}$. Assume further that: $$\lim_{x\rightarrow0}g(x)=+\infty,\qquad \lim_{x\rightarrow\infty}f(x)=\beta$$ for some $\beta\in\mathbb{R}$.
Under which conditions can we make the approximation $f(g(x))=\mathcal{O}(f(g_0(x)))$ near $0$?
What can I deduce from $(1)$ in relation to the order of the remainder from the approximation $f(g_0(x))$ $-$ if anything? In particular can we write that: $$f(g(x))=f(g_0(x))+\mathcal{O}(x)$$