Suppose that $f,g$ are functions $\mathbb{R} \rightarrow \mathbb{R}$, such that $f(x_0) = \alpha $ and $g(f(x_0)) = \beta$, and both $f,g$ admit Taylor polynomials to arbitrary approximation at $x_0, f(x_0)$, respectively. My question is about finding the Taylor approximations of $g \circ f$ by simply substituting the Taylor approximation of $f$ in that of $g$ - specifically, does this always work?
I know that, if $\alpha = 0$, we can do this (by an argument from Canuto and Tabacco's Mathematical Analysis 1, chapter 7.3). However, does the above strategy work regardless of what $\alpha, \beta$ might be? If not, what are the conditions on $\alpha, \beta$, and what do we do when the above doesn't work (aside from directly computing derivatives of the composition w/ Faa di Bruno's formula)?