Summary: The purpose of this work is the analysis of the solutions to the following problems related to the fractional \(p\)-Laplacian in a Lipschitzian bounded domain \(\Omega\subset \mathbb{R}^N\), \[ \begin{cases} -\int_{\mathbb{R}^N}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y^{\alpha p}}dy= f(x,u) \quad & x\in\Omega, \\ u=g(x)\quad & x\in\mathbb{R}^N\backslash\Omega, \end{cases} \] where \(\alpha\in (0, 1)\) and the exponent \(p\) goes to infinity. In particular we will analyze the cases:

(i)

\(f = f(x)\).

(ii)

\(f = f(u) = |u|^{\theta(p)-1}u\) with \(0 < \theta(p) < p- 1\) and \(\lim_{p\to\infty}\frac{\theta(p)}{p-1}=\Theta < 1\) with \(g\geq0\).

We show the convergence of the solutions to certain limit as \(p\) and identify the limit equation. In both cases, the limit problem is closely related to the Infinity Fractional Laplacian: \[ \mathcal{L}_\infty v(x) =\mathcal{L}^+_\infty v(x) + \mathcal{L}^-_\infty v(x), \] where \[ L^+_\infty v(x) = \displaystyle\sup_{y\in\mathbb{R}^N} \frac{v(y)- v(x)}{|y - x|^\alpha},\quad \mathcal{L}^-_\infty v(x) = \displaystyle\mathrm{inf}_{y\in\mathbb{R}^N} \frac{v(y)- v(x)}{|y - x|^\alpha}. \]