Scaling Limits of Polynomials and Entire Functions of Exponential Type D. S. Lubinsky ⋆ Abstract The connection between polynomials and entire functions of ex- ponential type is an old one, in some ways harking back to the simple limit lim n→∞  1+ z n  n = e z . On the left-hand side, we have P n ( z n ) , where P n is a polynomial of degree n, and on the right, an entire function of exponential type. We discuss the role of this type of scaling limit in a number of topics: Bernstein’s constant for ap- proximation of |x|; universality limits for random matrices; asymptotics of L p Christoffel functions and Nikolskii inequalities; and Marcinkiewicz-Zygmund inequalities. Along the way, we mention a number of unsolved problems. 1 Introduction The classical limit lim n→∞  1+ z n  n = e z , (1.1) plays a role in many areas of mathematics, expressing very simply the scaling limit of a sequence of a polynomials as an entire function of exponential type 1. Recall that an entire function f has exponential type A if for every ε> 0, |f (z )| = O  e (A+ε)|z|  , as |z |→∞, D. S. Lubinsky School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160 USA e-mail: lubinsky@math.gatech.edu ⋆ Research supported by NSF grant DMS1362208 1