We show that the solutions to the nonlocal obstacle problems for the nonlocal \(-\Delta _p^s\) operator, when the fractional parameter \(s\rightarrow \sigma \) for \(0<\sigma \le 1\) , converge to the solution of the corresponding obstacle problem for \(-\Delta _p^\sigma \) , being \(\sigma =1\) the classical obstacle problem for the local p-Laplacian. We discuss the weak stability of the quasi-characteristic functions of coincidence sets of the solution with the obstacle, which is a strong convergence of their characteristic functions when \(s\nearrow 1\) under a nondegeneracy condition. This stability can be shown also in terms of the convergence of the free boundaries, as well as of the coincidence sets, in Hausdorff distance when \(s\nearrow 1\) , under non-degeneracy local assumptions on the external force and a local topological property of the coincidence set of the limit classical obstacle problem for the local p-Laplacian, essentially when the limit coincidence set is the closure of its interior.
