We relate asymptotics of Jacobi parameters to asymptotics of the spectral weights near the edges. Typical of our results is that for a n ≡ 1, b n = −Cn −β (0 < β < 2 3 ), one has dµ(x) = w(x) dx on (−2, 2), and near x = 2, w(x) = e −2Q(x) where(1 + O((2 − x))).