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We consider infinite sequences {R_n} of successive local quadratic transformsof a regular local ring. Let S denote the directed union of the sequence ofregular local rings R_n. We previously showed the existence of a unique limitpoint V of the family of order valuation rings of the sequence. In this paper,we examine asymptotic properties of this family of order valuations. We linkthis asymptotic behavior to ring-theoretic properties of S, namely whether S isarchimedean and whether S is completely integrally closed. We constructexamples of such S that are archimedean and completely integrally closed butnot valuation domains. We give an explicit description of V, where thedescription depends on whether S is archimedean or non-archimedean.

Asymptotic properties of infinite directed unions of local quadratic transforms