Consider $T(x)= d \, x$ (mod 1) acting on $S^1$, a Lipschitz potential $A:S^1 \to \mathbb{R}$, $0<λ<1$ and the unique function $b_λ:S^1 \to \mathbb{R}$ satisfying $ b_λ(x) = \max_{T(y)=x} \{ λ\, b_λ(y) + A(y)\}.$ We will show that, when $λ\to 1$, the function $b_λ- \frac{m(A)}{1-λ}$ converges uniformly to the calibrated subaction $V(x) = \max_{μ\in \mathcal{ M}} \int S(y,x) \, d μ(y)$, where $S$ is the Mañe potential, $\mathcal{ M}$ is the set of invariant probabilities with support on the Aubry set and $m(A)= \sup_{μ\in \mathcal{M}} \int A\,dμ$. For $β>0$ and $λ\in (0,1)$, there exists a unique fixed point $u_{λ,β} :S^1\to \mathbb{R}$ for the equation $e^{u_{λ,β}(x)} = \sum_{T(y)=x}e^{βA(y) +λu_{λ,β}(y)}$. It is known that as $λ\to 1$ the family $e^{[u_{λ,β}- \sup u_{λ,β}]}$ converges uniformly to the main eigenfuntion $φ_β$ for the Ruelle operator associated to $βA$. We consider $λ=λ(β)$, $β(1-λ(β))\to+\infty$ and $λ(β) \to 1$, as $β\to\infty$. Under these hypothesis we will show that $\frac{1}β(u_{λ,β}-\frac{P(βA)}{1-λ})$ converges uniformly to the above $V$, as $β\to \infty$. The parameter $β$ represents the inverse of temperature in Statistical Mechanics and $β\to \infty$ means that we are considering that the temperature goes to zero. Under these conditions we get selection of subaction when $β\to \infty$.