The asymptotic behavior of solutions to a family of Dirichlet boundary value problems involving differential operators in divergence form on a domain equipped with a Finsler metric is investigated. Solutions are shown to converge uniformly to the distance function to the boundary of the domain which takes into account the Finsler norm involved in the equation. This implies that a well-known result in the analysis of problems modelling torsional creep continues to hold in this more general setting.