  We study the convergence of 1D Schr\"odinger ope\-rators $H_\varepsilon$ with
the potentials which are regularizations of a class of pseudo-potentials having
in particular the form $$ \alpha \delta'(x)+\beta
\delta(x)+\gamma/|x|\quad\text{or}\quad \alpha \delta'(x)+\beta
\delta(x)+\gamma/x. $$ The limit behaviour of $H_\varepsilon$ in the norm
resolvent topology, as $\varepsilon\to 0$, essentially depends on a way of
regularization of the Coulomb potential and the existence of zero-energy
resonances for $\delta'$-like potential. All possible limits are described in
terms of point interactions at the origin. As a consequence of the convergence
results, different kinds of $L^\infty(\mathbb{R})$-approximations to the even
and odd Coulomb potentials, both penetrable and impenetrable in the limit, are
constructed.
