  The homogeneous spectrum of a multigraded finitely generated algebra (in the
sense of Brenner-Schr\"oer) always admits an embedding into a toric variety
that is not necessarily separated, a so-called toric prevariety. In order to
have a convenient framework to study the tropicalization of homogeneous spectra
we propose a tropicalization procedure for toric prevarieties and study its
basic properties. With these tools at hand, we prove a generalization of
Payne's and Foster--Gross--Payne's tropical limit theorem for divisorial
schemes.
