  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 $\lambda$ and $\mu$, with $\lambda>\mu$, and an
absorption probability (at the origin) $\alpha\in(0,1]$. 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; $\mu\to\infty$, with
$\lambda=\beta\mu$ for some $\beta>1$ and $x>0$, in the second case. We prove
several large and moderate deviation results. We also present numerical
estimates of $\beta$ based on an asymptotic Normality result for the case of
the second scaling.
