abstract: Let $u$ be a (continuous) function whose first order distributional gradient is $p$-integrable with $p\ge 1$ strictly less than the dimension of the underlying Euclidean space. Then there is a real number $c$ so that $\lim{t\to \infty}u(tw)=c$ for almost every $w\in S{n-1},$ the boundary of the unit ball. When $p$ is larger or equal to the dimension, this conclusion may fail even if $u$ is additionally bounded. I will describe our (joint work with Josh Kline and Khanh Nguyen) attemps towards establishing metric versions of these statements.