  We study a class of delta-like perturbations of the Laplacian on the
half-line, characterized by Robin boundary conditions at the origin. Using the
formalism of nonstandard analysis, we derive a simple connection with a
suitable family of Schr\"{o}dinger operators with potentials of very large
(infinite) magnitude and very short (infinitesimal) range. As a consequence, we
also derive a similar result for point interactions in the Euclidean space
$\mathbb{R}^3$, in the case of radial potentials. Moreover, we discuss
explicitly our results in the case of potentials that are linear in a
neighbourhood of the origin.
