  Several semilocal exchange potentials usually employed in the framework of
density-functional theory (DFT) are tested and compared with their exact
counterpart, the exchange Optimized Effective Potential (OEP), as applied to
the jellium-slab model of a metal-vacuum interface. Driven by their explicit
dependence on the ground-state density, its gradient, and its kinetic-energy
density, the three analyzed semilocal exchange potentials approach their
respective asymptotic limits faster than in the case of the OEP, all of them
having an asymptotic scaling of the form $-\alpha\,e^2/z + V_{\infty}$, with
$\alpha < 1$. Here we provide the leading analytic asymptotics of the three
model potentials under study, and we find that none of them exhibits the exact
OEP slab asymptotics $-\;e^2/z$. While the so-called Becke-Roussel potential's
leading asymptote is close to its exact OEP counterpart, the other two model
potentials under study approach a material-dependent positive constant value
far into the vacuum, resulting in considerably overestimated ionization
potentials.
