  In this paper, we study the existence of limits at infinity along almost
every infinite curve for the upper and lower approximate limits of bounded
variation functions on complete unbounded metric measure spaces. We prove that
if the measure is doubling and supports a $1$-Poincar\'e inequality, then for
every bounded variation function $f$ and for $1$-a.e. infinite curve $\gamma$,
for both the upper approximate limit $f^\vee$ and the lower approximate limit
$f^\wedge$ we have that \[ \lim_{t\to+\infty}f^\vee(\gamma(t)) {\rm \ \ and\ \
}\lim_{t\to+\infty}f^\wedge(\gamma(t)) \] exist and are equal to the same
finite value. We give examples showing that the conditions of doubling and a
$1$-Poincar\'e inequality are also necessary for the existence of limits.
Furthermore, we establish a characterization for strictly positive $1$-modulus
of the family of all infinite curves in terms of bounded variation functions.
These generalize results for Sobolev functions given in \cite{KN23}.
