Let $(R,m)$ be a complete equicharacteristic noetherian local domain and $\nu$ a valuation of its field of fractions whose valuation ring $R_\nu$ dominates $R$ with trivial residue field extension $k\simeq k_\nu$. In order to extend to all rational valuations the torific method (se below) of local uniformization which works for rational valuations with finitely generated semigroup, new methods are needed. One such method is to approximate the given valuation by semivaluations (valuations on quotients, also called pseudo-valuations) with finitely generated semigroups. We produce equations in a generalized power series ring for the algebra encoding the degeneration. We use this to represent $\nu$ as the limit of a sequence of semivaluations of $R$ with finitely generated semigroups.