  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 epsilon multiplicity of reduced standard
graded algebras over an excellent local ring exists as a limit. We also obtain
some important special cases of Cutkosky's results concerning
$\varepsilon$-multiplicity, as corollaries of our main theorem.
