In this article we establish the asymptotic behavior of generating functions related to the exponential sum over finite fields of elementary symmetric functions and their perturbations. This asymptotic behavior allows us to calculate the probability generating function of the probability that the the elementary symmetric polynomial of degree \(k\) and its perturbations returns \(\beta \in \mathbb{F}_q\) where \(\mathbb{F}_q\) represents the field of \(q\) elements.
