  We study the $\Gamma$-convergence of the functionals $F_n(u):= ||
f(\cdot,u(\cdot),Du(\cdot))||_{p_n(\cdot)}$ and $\mathcal{F}_n(u):=
\int_{\Omega} \frac{1}{p_n(x)} f^{p_n(x)}(x,u(x),Du(x))dx$ defined on $X\in
\{L^1(\Omega,\mathbb{R}^d), L^\infty(\Omega,\mathbb{R}^d),
C(\Omega,\mathbb{R}^d)\}$ (endowed with their usual norms) with effective
domain the Sobolev space $W^{1,p_n(\cdot)}(\Omega, \mathbb{R}^d )$. Here
$\Omega\subseteq \mathbb{R}^N$ is a bounded open set, $N,d \ge 1$ and the
measurable functions $p_n: \overline{\Omega} \rightarrow (1, + \infty) $
satisfy the conditions ${\mathop{\rm ess\: sup }}_{\ \overline \Omega} p_n \le
\, \beta \, {\mathop{\rm ess\: inf }}_{\ \overline \Omega} p_n $ for a fixed
constant $\beta > 1$ and $ {\mathop{\rm ess\: inf }}_{\ \overline \Omega} p_n
\rightarrow + \infty$ as $n \rightarrow + \infty$. We show that when
$f(x,u,\cdot)$ is level convex and lower semicontinuous and it satisfies a
uniform growth condition from below, then, as $n\to \infty$, the sequences
$(F_n)_n$ $\Gamma$-converges in $X$ to the functional $F$ represented as $F(u)=
|| f(\cdot,u(\cdot),Du(\cdot))||_{\infty}$ on the effective domain
$W^{1,\infty}(\Omega, \mathbb{R}^d )$. Moreover we show that the
$\Gamma$-$\lim_n \mathcal F_n$ is given by the functional $
\mathcal{F}(u):=\left\{\begin {array}{lll} \!\!\!\!\!\! & 0 & \hbox{if } ||
f(\cdot,u(\cdot),Du(\cdot)) ||_{\infty}\leq 1,\\ \!\!\!\!\!\! & +\infty &
\hbox{otherwise in } X.\\ \end{array}\right. $
