We present an asymptotic analysis of adaptive methods for L^p approximation of functions f ∈ C^r([a, b]), where . The methods rely on piecewise polynomial interpolation of degree r − 1 with adaptive strategy of selecting m subintervals. The optimal speed of convergence is in this case of order m^−r and it is already achieved by the uniform (nonadaptive) subdivision of the initial interval; however, the asymptotic constant crucially depends on the chosen strategy. We derive asymptotically best adaptive strategies and show their applicability to automatic L^p approximation with a given accuracy ε.