We describe approximation algorithms for (unweighted) MAX SAT with performance ratios arbitrarily close to 1, in particular, when performance ratios exceed the limits of polynomial-time approximation. Namely, given a polynomial-time α-approximation algorithm A₀, we construct an (α+ε)-approximation algorithm A. The algorithm A runs in time of the order c^εk, where k is the number of clauses in the input formula and c is a constant depending on α. Thus we estimate the cost of improving a performance ratio. Similar constructions for MAX 2SAT and MAX 3SAT are also described. Taking known algorithms as A₀ (for example, the Karloff-Zwick algorithm for MAX 3SAT), we obtain particular upper bounds on the running time of A.