We consider a Sobolev inner product such as (1)(f,g) S =∫f(x)g(x)dμ 0 (x)+λ∫f'(x)g'(x)dμ 1 (x),λ>0,with (μ 0 ,μ 1 ) being a symmetrically coherent pair of measures with unbounded support. Denote by Q n the orthogonal polynomials with respect to (1) and they are so-called Hermite-Sobolev orthogonal polynomials. We give a Mehler-Heine-type formula for Q n when μ 1 is the measure corresponding to Hermite weight on R, that is, dμ 1 =e - x 2 dx and as a consequence an asymptotic property of both the zeros and critical points of Q n is obtained, illustrated by numerical examples. Some remarks and numerical experiments are carried out for dμ 0 =e - x 2 dx. An upper bound for |Q n | on R is also provided in both cases.