  We investigate asymptotic polynomial approximation for a class of weighted
Bloch functions in the unit disc. Our main result is a structural theorem on
asymptotic polynomial approximation in the unit disc, in the flavor of the
classical Plessner Theorem on asymptotic values of meromorphic functions. This
provides the appropriate set up for studying metric and geometric properties of
sets E on the unit circle for which the following simultaneous approximation
phenomenon occurs: there exists analytic polynomials which converge uniformly
to zero on E and to a non-zero function in the weighted Bloch norm. We offer a
characterization completely within the realm of real-analysis, establish a
connection to removable sets for analytic Sobolev functions in the complex
plane, and provide several necessary conditions in terms of entropy, Hausdorff
content and condenser capacity. Furthermore, we demonstrate two principal
applications of our developments, which go in different directions. First, we
shall deduce a rather subtle consequence in the theme of smooth approximation
in de Branges-Rovnyak spaces. Secondly, we answer some questions that were
raised almost a decade ago in the theory of Universal Taylor series.
