In this paper, we study the relationship between p-harmonic functions and absolutely minimizing Lipschitz extensions in the setting of a metric measure space (X, d, μ). In particular, we show that limits of p-harmonic functions (as p → ∞) are necessarily the ∞-energy minimizers among the class of all Lipschitz functions with the same boundary data. Our research is motivated by the observation that while the p-harmonic functions in general depend on the underlying measure μ, in many cases their asymptotic limit as p → ∞ turns out have a characterization that is independent of the measure.