  The k-limited packing number, $L_k(G)$, of a graph $G$, introduced by
Gallant, Gunther, Hartnell, and Rall, is the maximum cardinality of a set $X$
of vertices of $G$ such that every vertex of $G$ has at most $k$ elements of
$X$ in its closed neighbourhood. The main aim in this paper is to prove the
best-possible result that if $G$ is a cubic graph, then $L_2(G) \geq |V
(G)|/3$, improving the previous lower bound given by Gallant, \emph{et al.}
  In addition, we construct an infinite family of graphs to show that lower
bounds given by Gagarin and Zverovich are asymptotically best-possible, up to a
constant factor, when $k$ is fixed and $\Delta(G)$ tends to infinity. For
$\Delta(G)$ tending to infinity and $k$ tending to infinity sufficiently
quickly, we give an asymptotically best-possible lower bound for $L_k(G)$,
improving previous bounds.
