  There is a vast theory of Chebyshev and residual polynomials and their
asymptotic behavior. The former ones maximize the leading coefficient and the
latter ones maximize the point evaluation with respect to an $L^\infty$ norm.
We study Chebyshev and residual extremal problems for rational functions with
real poles with respect to subsets of $\overline{\mathbb{R}}$. We prove root
asymptotics under fairly general assumptions on the sequence of poles.
Moreover, we prove Szeg\H{o}--Widom asymptotics for sets which are regular for
the Dirichlet problem and obey the Parreau--Widom and DCT conditions.
