For a closed Riemannian manifold (Mm, g) of constant positive scalar curvature and any other closed Riemannian manifold (Nn, h), we show that the limit of the Yamabe constants of the Riemannian products (M × N, g + rh) as r goes to infinity is equal to the Yamabe constant of (Mm × Rn, [g + gE ]) and is strictly less than the Yamabe invariant of Sm+n provided n ≥ 2. We then consider the minimum of the Yamabe functional restricted to functions of the second variable and we compute the limit in terms of the best constants of the Gagliardo–Nirenberg inequalities.