  Consider a topologically transitive countable Markov shift and, let $f$ be a
summable potential with bounded variation and finite Gurevic pressure. We prove
that there exists an equilibrium state $\mu_{tf}$ for each $t > 1$ and that
there exists accumulation points for the family $(\mu_{tf})_{t>1}$ as $t \to
\infty$. We also prove that the Kolmogorov-Sinai entropy is continuous at
$\infty$ with respect to the parameter $t$, that is $\lim_{t \to \infty}
h(\mu_{tf})=h(\mu_{\infty})$, where $\mu_{\infty}$ is an accumulation point of
the family $(\mu_{tf})_{t>1}$. These results do not depend on the existence of
Gibbs measures and, therefore, they extend results of \cite{MaUr01} and
\cite{Sar99} for the existence of equilibrium states without the BIP property,
\cite{JMU05} for the existence of accumulation points in this case and,
finally, we extend completely the result of \cite{Mor07} for the entropy zero
temperature limit beyond the finitely primitive case.
