ASYMPTOTIC FUNCTIONS

In this paper, we improve some of results in [2] by showing that if I is a cnacellation ideal and if J is a regular ideal then $\alpha$(m), $\beta$(m) and $\delta$(m), behave nicely under localization. We prove that lim \ulcorner=0 if and only if $\alpha$(m) is eventually constant and that lim\ulcorner exists and is equal to or less than $\alpha$(1). Finally we give several conditions which are equivalent to $lim_{m{\rightarrow}{\infty}}{\frac{{\alpha}(m)}{m}}=0$.