Epsilon multiplicity for Noetherian graded algebras

The notion of $\varepsilon$-multiplicity was originally defined by Ulrich and Validashti as a limsup and they used it to detect integral dependence of modules. It is important to know if it can be realized as a limit. In this article we show that the relative $\varepsilon$-multiplicity of reduced Noetherian graded algebras over an excellent local ring exists as a limit. We also obtain a generalization of Cutkosky's result concerning $\varepsilon$-multiplicity, as a corollary of our main theorem.