We consider the so-called Nevai–Sobolev orthogonal polynomials, which are orthogonal with respect to a Sobolev inner product involving the Freud weight \(\exp (-x^4).\) Our aim is to study the local asymptotics, known as Mehler–Heine asymptotics, for these polynomials. As a consequence, this type of asymptotics allows us to deduce the asymptotic behavior of the corresponding scaled zeros and critical points. At this point, we address the problem of computing the zeros of this family of polynomials. In fact, we transform this problem into solving a generalized eigenvalue problem which is constructed from a recurrence relation satisfied by the Nevai–Sobolev orthogonal polynomials. Besides, how to handle commercial software to perform these computations will be a topic of interest and we will show different numerical examples illustrating the computation of the zeros. Finally, the fourth-order differential equation satisfied by these orthogonal polynomials is deduced.
