Asymptotic elasticity in atomic monoids

Let M be a commutative atomic monoid (i.e. every nonzero nonunit of M can be factored as a product of irreducible elements). Let p(x) denote the elasticity of x is an element of M; R(M)={rho(x) vertical bar x is an element of M} the set of elasticities of elements in M, and rho(M) = sup R(M) the elasticity of M. Define (rho) over bar (x)=lim(n ->infinity)(x(n)) to be the asymptotic elasticity of x. We determine some basic properties of the function (p) over bar and determine its image for certain block monoids.