  We consider the limit of sequences of normalized $(s,2)$-Gagliardo seminorms
with an oscillating coefficient as $s\to 1$. In a seminal paper by Bourgain,
Brezis and Mironescu (subsequently extended by Ponce) it is proven that if the
coefficient is constant then this sequence $\Gamma$-converges to a multiple of
the Dirichlet integral. Here we prove that, if we denote by $\varepsilon$ the
scale of the oscillations and we assume that $1-s<\!<\varepsilon^2$, this
sequence converges to the homogenized functional formally obtained by
separating the effects of $s$ and $\varepsilon$; that is, by the homogenization
as $\varepsilon\to 0$ of the Dirichlet integral with oscillating coefficient
obtained by formally letting $s\to 1$ first.
