In this paper, we study the behaviour at infinity of p-Sobolev functions in the setting of Ahlfors Q-regular metric measure spaces supporting a p-Poincaré inequality. By introducing the notions of sets which are p-thin at infinity, we show that functions in the homogeneous space \(\dot{N}^{1,p}(X)\) necessarily have limits at infinity outside of p-thin sets, when \(1\le p<Q<+\infty \) . When \(p>Q\) , we show by example that uniqueness of limits at infinity may fail for functions in \(\dot{N}^{1,p}(X)\) . While functions in \(\dot{N}^{1,p}(X)\) may not have any reasonable limit at infinity when \(p=Q\) , we introduce the notion of a Q-thick set at infinity, and characterize the limits of functions in \(\dot{N}^{1,Q}(X)\) along infinite curves in terms of limits outside Q-thin sets and along Q-thick sets. By weakening the notion of a thick set, we show that a function in \(\dot{N}^{1,Q}(X)\) with a limit along such an almost thick set may fail to have a limit along any infinite curve. While homogeneous p-Sobolev functions may have infinite limits at infinity when \(p\ge Q\) , we provide bounds on how quickly such functions may grow: when \(p=Q\) , functions in \(\dot{N}^{1,p}(X)\) have sub-logarithmic growth at infinity, whereas when \(p>Q\) , such functions have growth at infinity controlled by \(d(\cdot , O)^{1-Q/p}\) , where O is a fixed base point in X. For the inhomogeneous spaces \(N^{1,p}(X)\) , the phenomenon is different. We show that for \(1\le p\le Q\) , the limit of a function \(u\in N^{1,p}(X)\) is zero outside of a p-thin set, whereas \(\lim _{x\rightarrow +\infty }u(x)=0\) for all \(u\in N^{1,p}(X)\) when \(p>Q\) .
