We will consider the scaling limits of polynomial reproducing kernels for measures on the real line. For many years there has been considerable research to find the weakest assumptions that one can place on a measure that allows one to prove that these rescaled kernels converge to the sinc kernel. Our main result will provide the weakest conditions that have yet been found. In particular, it will demonstrate that one only needs local conditions on the measure. We will also settle a conjecture of Avila, Last, and Simon by showing that convergence holds at almost every point in the essential support of the absolutely continuous part of the measure. Speaker(s): Brian Simanek (Baylor University)