  For $k \geq 1$ and a graph $G$ without isolated vertices, a \emph{total
(distance) $k$-dominating set} of $G$ is a set of vertices $S \subseteq V(G)$
such that every vertex in $G$ is within distance $k$ to some vertex of $S$
other than itself. The \emph{total (distance) $k$-domination number} of $G$ is
the minimum cardinality of a total $k$-dominating set in $G$, and is denoted by
$\gamma_{k}^t(G)$. When $k=1$, the total $k$-domination number reduces to the
\emph{total domination number}, written $\gamma_t(G)$; that is, $\gamma_t(G) =
\gamma_{1}^t(G)$. This paper shows that several known lower bounds on the total
domination number generalize nicely to lower bounds on total (distance)
$k$-domination.
