Limits op groupoid C^*-algebras arising from open covers

I. Raeburn and J. Taylor have constructed continuous-trace C ^*-algebras with a prescribed Dixmier-Douady class, which also depend on the choice of a locally finite open cover of the spectrum. We study the asymptotic behavior of these algebras with respect to certain refinements of the cover and appropriate extension of cocycles. This leads to the analysis of a limit groupoid G and a cocycle σ, and the algebra C^*(G, σ) may be regarded as a generalized direct limit of the Raeburn-Taylor algebras. As a special case, all UHF C^*-algebras arise from this limit construction.