We study the boundary layer effects in the small relaxation limit to the equilibrium scalar conservation laws in 1-space dimension. More precisely, we consider the following initial boundary value problem of hyperbolic system with relaxation \begin{equation} \begin{array}{rcl} && u^{\varepsilon}_t + v^{\varepsilon}_x = 0 \\ && v^{\varepsilon}_t + \alpha ^2 u^{\varepsilon}_x = \frac{1}{\varepsilon} (v^{\varepsilon} - f(u^{\varepsilon})), \quad e> 0, x>0,t>0, u^{\varepsilon} , v^{\varepsilon} \in {\bf \it R} \\ && u^{\varepsilon} (x,0) = u_0 (x), v^{\varepsilon} (x,0) = v_0 (x), \quad x>0 \\ && u^{\varepsilon} (0,t) = u_b (t), \quad t>0 \end{array} \end{equation} \noindent with the data $u_0, u_v$ and $u_b$ in $BV$ . First, it is shown that for initial and boundary data satisfying a strict version of sub-characteristic condition, there exists unique global (in time) solution, $(u^{\varepsilon}, v^{\varepsilon})$, to ( 1) for each $\varepsilon > 0$. The spatial total variation of $(u^{\varepsilon}, v^{\varepsilon})$ is shown to be bounded independent of $\varepsilon$, and consequently, a subsequence of $(u^{\varepsilon}, v^{\varepsilon})$ is bounded independent of $\varepsilon$, and a subsequence of $(u^{\varepsilon}, v^{\varepsilon})$ converges to a limit $(u, f(u))$ as $\varepsilon \rightarrow 0^+$. Furthermore, $u(x,t)$ is a weak solution to the scalar conservation law \begin{equation} \begin{array}{rcl} && u_t + f(u)_x =0, \quad x>0, t>0, \\ && u(x,0) = u_0 (x), \quad x>0. \end{array} \end{equation} Next, we prove that for data which are suitably small perturbations of a nontransonic state $(f' \neq 0)$, the relaxation limit function statisfies an boundary-entropy condition of the form \begin{equation} \int^T_0 \int_{\Omega} \phi ^k (u) \rho _t + \psi ^k (u) \rho _{\psi} \geq \int ^T_0 \int _{\partial \Omega} \Psi ^k (u_b, f (\gamma \circ u)) \rho \cdot {\bf \it n}, \end{equation} \noindent for all $k \in {\bf \it R}$ and all nonnegative test function $\rho \in C^{\infty}_u (\Omega \times (0,T))$. Where $\Omega = \{ x > 0 \}, \gamma\circ u$ is the trace of $u$ on $\partial \Omega \times (0,T)$, {\bf \it n} the outer normal, \[ (\phi ^k (u), \psi ^k (u))=(|u-k|, {sgn} (u-k)(f(u)-f(k))\] the Kurzkov's entropy-flux pair and $\Psi ^k$ the corresponding derived entropy flux for the relaxation system (1). Finally, the weak solutions to (2) with the boundary-entropy condition (3) is shown to be unique. Consequently, the relaxation limit of solutions to (1) is unique and the whole sequence converges to the unique limit. One consequence of our analysis shows that the boundary layers occur only in the $u$-components in the sense that $v^{\varepsilon} (0, \cdot )$ converges strongly to $\gamma \circ v = f (\gamma \circ u)$.