This work investigates a limit problem for an anisotropic p-Laplacian operator as \( p \rightarrow \infty \) within the framework of viscosity solutions. Specifically, we analyze the asymptotic behavior of an eigenvalue problem subject to Robin and mixed boundary conditions: \( \left\{ \begin{aligned} -\operatorname {div} {\mathscr {H}}_{p}(\nabla u)&= \Lambda _p |u|^{p-2} u & \text {in } \Omega , \\ {\mathscr {H}}_{p}(\nabla u) \cdot \nu + \beta ^p |u|^{p-2} u&= {\Lambda _p |u|^{p-2} u} & \text {on } \partial \Omega . \end{aligned} \right. \) We demonstrate that the limit of the eigenfunction is a viscosity solution to an eigenvalue problem governed by an anisotropic \( \infty \) -Laplacian, and we establish several geometric properties of the corresponding eigenvalues. Subsequently, utilizing the eigenvalues derived in the first part, we address the problem with a forcing term \( f \in L^\infty \) and a boundary condition \( g \in L^\infty \) . Furthermore, we investigate the asymptotic behavior of the associated solutions as \( p \rightarrow \infty \) : \( \left\{ \begin{aligned} -\operatorname {div} {\mathscr {H}}_{p}(\nabla u)&= f & \text {in } \Omega , \\ {\mathscr {H}}_{p}(\nabla u) \cdot \nu + \beta ^p |v|^{p-2} v&= g & \text {on } \partial \Omega . \end{aligned} \right. \)
