  In this paper we firstly study the limit of minimizers of the fractional
$W^{s,p}$-norms as $p\rightarrow+\infty$ in De Giorgi sense. In particular, we
analyzed the $\Gamma$-convergence of non-homogeneous Dirichlet boundary problem
for fractional $p$-Laplacian in this approximation process, and proved that as
$p\rightarrow+\infty$ the minimizers of fractional $p$-Laplacian with Dirichlet
boundary $\Gamma$-converges to a minimizer of H\"{o}lder $\infty$-Laplacian
under the same Dirichlet boundary condition. On the other hand, we first
investigate the asymptotic behaviour of non-homogeneous fractional
$p$-functionals when $k\rightarrow s$ from above; then we study the
approximation process as $k\rightarrow s$ from below of a free fractional
$p$-functional, during which we will find some special phenomenon different
from the case from above. Both of the way to dispose these two asymptotic
directions are in the De Giorgi sense.
