We study limits at infinity for homogeneous Hajlasz-Sobolev functions defined on uniformly perfect metric spaces equipped with a doubling measure. We prove that a quasicontinuous representative of such a function has a pointwise limit at infinity outside an exceptional set, defined in terms of a variational relative capacity. Our framework refines earlier approaches that relied on Hausdorff content rather than relative capacity, and it extends previous results for homogeneous Newtonian and fractional Sobolev functions.
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