In this manuscript, we focus on studying a family of viscosity solutions \((u_\varepsilon )_{\varepsilon > 0}\) for a singular perturbation problem driven by the normalized \(p(x)\) -Laplacian operator \( \left\{ \begin{array}{rclcl} \Delta _{p_{\varepsilon } (x)}^{\textrm{N}} u_{\varepsilon }(x) & = & \zeta _{\varepsilon }\left( u_{\varepsilon }\right) + f_{\varepsilon }(x) & {\text {in}} & \Omega , \\ u_{\varepsilon }(x) & = & g(x) & {\text {on}} & \partial \Omega , \end{array} \right. \) We establish that the solutions exhibit uniform bounds, local Lipschitz continuity, and non-degeneracy properties in a regular domain \(\Omega \subset {\mathbb {R}}^n\) with a sufficiently smooth boundary datum. As a consequence, we demonstrate that, up to a subsequence, \( \lim _{j \rightarrow \infty } u_{\varepsilon _j} = u_0\) , where \(u_0\) is a viscosity solution to a one-phase Bernoulli-type free boundary problem, enjoying uniform and optimal Lipschitz bounds.
