Suppose I have a analytic function $f : \mathbb{R} \to \mathbb{R}$ and I have the asymptotic expansion of some $x_0$ up to a few terms in terms of $\epsilon$ for some $\epsilon \to 0$ which I believe is such that $f(x_0) = 0$. How can I prove that $$f(x_0) = 0$$
Some context - In my case, I am trying to compute the asymptotic expansion of the solution to some saddle point equation of form $f(x) = 0$, in order to apply the saddle point method from Asymptotic Analysis/Analytic combinatorics/analytic number theory. However, actually computing the solution to this equation seems difficult. Instead, I want to 'guess' the solution to the equation, and prove that my guess is correct. Is this possible?