  Let $\Omega\subset\mathbb{R}^n$ be a bounded $(\varepsilon,\infty)$-domain
with $\varepsilon\in(0,1]$, $X(\mathbb{R}^n)$ a ball Banach function space
satisfying some extra mild assumptions, and $\{\rho_\nu\}_{\nu\in(0,\nu_0)}$
with $\nu_0\in(0,\infty)$ a $\nu_0$-radial decreasing approximation of the
identity on $\mathbb{R}^n$. In this article, the authors establish two
extension theorems, respectively, on the inhomogeneous ball Banach Sobolev
space $W^{m,X}(\Omega)$ and the homogeneous ball Banach Sobolev space
$\dot{W}^{m,X}(\Omega)$ for any $m\in\mathbb{N}$. On the other hand, the
authors prove that, for any $f\in\dot{W}^{1,X}(\Omega)$, $$ \lim_{\nu\to0^+}
\left\|\left[\int_\Omega\frac{|f(\cdot)-f(y)|^p}{
|\cdot-y|^p}\rho_\nu(|\cdot-y|)\,dy \right]^\frac{1}{p}\right\|_{X(\Omega)}^p
=\frac{2\pi^{\frac{n-1}{2}}\Gamma (\frac{p+1}{2})}{\Gamma(\frac{p+n}{2})}
\left\|\,\left|\nabla f\right|\,\right\|_{X(\Omega)}^p, $$ where $\Gamma$ is
the Gamma function and $p\in[1,\infty)$ is related to $X(\mathbb{R}^n)$. Using
this asymptotics, the authors further establish a characterization of
$W^{1,X}(\Omega)$ in terms of the above limit. To achieve these, the authors
develop a machinery via using a method of the extrapolation, two extension
theorems on weighted Sobolev spaces, and some recently found profound
properties of $W^{1,X}(\mathbb{R}^n)$ to overcome those difficulties caused by
that the norm of $X(\mathbb{R}^n)$ has no explicit expression and that
$X(\mathbb{R}^n)$ might be neither the reflection invariance nor the
translation invariance. This characterization has a wide range of generality
and can be applied to various Sobolev-type spaces, such as Morrey
[Bourgain--Morrey-type, weighted (or mixed-norm or variable), local (or global)
generalized Herz, Lorentz, and Orlicz (or Orlicz-slice)] Sobolev spaces, all of
which are new.
