We explain how to find the asymptotic form of fixed point solutions in functional truncations, in particular $f(R)$ approximations. We find that quantum fluctuations do not decouple at large $R$, typically leading to elaborate asymptotic solutions containing several free parameters. By a counting argument, these can be used to map out the dimension of the fixed point solution spaces. They are also necessary to validate the numerical solution, and provide the physical part in the limit that the cutoff is removed: the fixed point equation of state. As an example we apply the techniques to a recent $f(R)$ approximation by Demmel et al, finding asymptotic matches to their numerical solution. Depending on the value of the endomorphism parameter, we find many other asymptotic solutions and fixed point solution spaces of differing dimensions, yielding several alternative scenarios for the equation of state. Asymptotic studies of other $f(R)$ approximations are needed to clarify the picture.

Minor corrections. Version to be published in PRD