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We study the 
$R_\beta$-positivity and the existence of zero-temperature limits for a sequence of infinite-volume Gibbs measures 
$(\mu_{\beta}(\!\cdot\!))_{\beta \geq 0}$ at inverse temperature 
$\beta$ associated to a family of nearest-neighbor matrices 
$(Q_{\beta})_{\beta \geq 0}$ reflected at the origin. We use a probabilistic approach based on the continued fraction theory previously introduced in Ferrari and Martínez (1993) and sharpened in Littin and Martínez (2010). Some necessary and sufficient conditions are provided to ensure (i) the existence of a unique infinite-volume Gibbs measure for large but finite values of 
$\beta$, and (ii) the existence of weak limits as 
$\beta \to \infty$. Some application examples are revised to put in context the main results of this work.
