Abstract. Define S(n, β) to be the set of complex polynomials of degree n ≥ 2 with all roots in the unit disk and at least one root at β. For a polynomial P , define |P | β to be the distance between β and the closest root of the derivative P . Finally, define rn(β) = sup{|P | β : P ∈ S(n, β)}. In this notation, a conjecture of Bl. Sendov claims that rn(β) ≤ 1.In this paper we investigate Sendov's conjecture near the unit circle, by computing constants C 1 and C 2 (depending only on n) such that rn(β) ∼ 1 + C 1 (1 − |β|) + C 2 (1 − |β|) 2 for |β| near 1. We also consider some consequences of this approximation, including a hint of where one might look for a counterexample to Sendov's conjecture.