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We consider a varying discrete Sobolev inner product involving the Laguerre weight. Our aim is to study the asymptotic properties of the corresponding orthogonal polynomials and of their zeros. We are interested in Mehler-Heine type formulas because they describe the asymptotic differences between these Sobolev orthogonal polynomials and the classical Laguerre polynomials. Moreover, they give us an approximation of the zeros of the Sobolev polynomials in terms of the zeros of other special functions. We generalize some results appeared very recently in the literature for both the varying and non-varying cases.