We consider a telegraph process with elastic boundary at the origin studied recently in the literature. It is a particular random motion with finite velocity which starts at $x\geq 0$, and its dynamics is determined by upward and downward switching rates $λ$ and $μ$, with $λ>μ$, and an absorption probability (at the origin) $α\in(0,1]$. Our aim is to study the asymptotic behavior of the absorption time at the origin with respect to two different scalings: $x\to\infty$ in the first case; $μ\to\infty$, with $λ=βμ$ for some $β>1$ and $x>0$, in the second case. We prove several large and moderate deviation results. We also present numerical estimates of $β$ based on an asymptotic Normality result for the case of the second scaling.