Discriminant, vertex and general type of a family of polynomials

In this paper we give a formal expression to the limits of dual plane curves by using the method introduced by Katz. As an application, we give a formula to compute the vertices of $C(0)$ a plane curve in $C(t)$ a one-parameter family of plane curves and we show that those vertices depend only on the first $\tau$ terms of that family, where $\tau$ is the general type of $C(t)$.