In this paper, we consider the asymptotic behavior of the sequence of monic polynomials orthogonal with respect to the Sobolev inner product 〈p,q〉S=∫0∞p(x)q(x)dμ+Mp(m)(ζ)q(m)(ζ), where ζ < 0, M ⩾ 0 and dμ = e−xxαdx. We study the outer relative asymptotics of these polynomials with respect to the classical Laguerre polynomials, and we deduce a Mehler–Heine type formula and a Plancherel–Rotach type formula for the rescaled polynomials.