  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.
