  We establish the existence and uniqueness of limits at infinity along
infinite curves outside a zero modulus family for functions in a homogeneous
Sobolev space under the assumption that the underlying space is equipped with a
doubling measure which supports a Poincar\'e inequality. We also characterize
the settings where this conclusion is nontrivial. Secondly, we introduce
notions of weak polar coordinate systems and radial curves on metric measure
spaces. Then sufficient and necessary conditions for existence of radial limits
are given. As a consequence, we characterize the existence of radial limits in
certain concrete settings.
