We revisit the Bourgain-Brezis-Mironescu result that the Gagliardo-Norm of the fractional Sobolev space $W^{s,p}$, up to rescaling, converges to $W^{1,p}$ as $s\to 1$.

We do so from the perspective of Triebel-Lizorkin spaces, by finding sharp $s$-dependencies for several embeddings between $W^{s,p}$ and $F^{s,p}_q$ where $q$ is either $2$ or $p$.

We recover known results, find a few new estimates, and discuss some open questions.

Joint work with Denis Brazke, Po-Lam Yung.