  For a closed Riemannian manifold $(M^m,g)$ of constant positive scalar
curvature and any other closed Riemannian manifold $(N^n,h)$, we show that the
limit of the Yamabe constants of the Riemannian products $(M\times N,g+rh)$ as
$r$ goes to infinity is equal to the Yamabe constant of $(M^m \times R^n,
[g+g_E])$ and is strictly less than the Yamabe invariant of $S^{m+n}$ provided
$n\geq 2$. 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.
