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 Israel J. Math. 125 (2001), pp. 93-130 and Ergodic Theory Dynam. Systems 19 (1999), pp. 1565-1593 for the existence of equilibrium states without the big images and preimages (BIP) property, J. Stat. Phys. 119 (2005), pp. 765-776 for the existence of accumulation points in this case and, finally, we extend completely the result of J. Stat. Phys. 126 (2007), pp. 315-324 for the entropy zero temperature limit beyond the finitely primitive case.