  Let $\Omega$ be a smooth, bounded domain of $\mathbb{R}^{N}$, $\omega$ be a
positive, $L^{1}$-normalized function, and $0<s<1<p.$ We study the asymptotic
behavior, as $p\rightarrow\infty,$ of the pair $\left( \sqrt[p]{\Lambda_{p}%
},u_{p}\right) ,$ where $\Lambda_{p}$ is the best constant $C$ in the Sobolev
type inequality \[ C\exp\left( \int_{\Omega}(\log\left\vert u\right\vert
^{p})\omega \mathrm{d}x\right) \leq\left[ u\right] _{s,p}^{p}\quad\forall\,u\in
W_{0}^{s,p}(\Omega) \] and $u_{p}$ is the positive, suitably normalized
extremal function corresponding to $\Lambda_{p}$. We show that the limit pairs
are closely related to the problem of minimizing the quotient $\left\vert
u\right\vert _{s}/\exp\left( \int_{\Omega}(\log\left\vert u\right\vert )\omega
\mathrm{d}x\right) ,$ where $\left\vert u\right\vert _{s}$ denotes the
$s$-H\"{o}lder seminorm of a function $u\in C_{0}^{0,s}(\overline{\Omega}).$
