In this paper, we focus on four weights ω(z,s) = zλe−N(z+s(z2−z)), where z ∈ (0,∞), λ > −1, 0 ≤ s ≤ 1, N > 0; ω(z, t) = zλe−z2+tz, where z ∈ (0,∞), λ > −1, t ∈ R; ω(z, t1) = e−z2 (A + Bθ (z − t1)), with z ∈ R, A ≥ 0, A + B ≥ 0, B = 0, where θ (z) is the Heaviside step function; and ω(z) = |z| αe−N(z2+s(z4−z2)), with z ∈ R, α > −1, N > 0, 0 ≤ s ≤ 1. The second-order differential equations satisfied by Pn(z), the degree-n polynomials orthogonal with respect to each of these weights, are shown to be asymptotically equivalent to the bi-confluent Heun equations as n → ∞. In most cases, a parameter other than n must simultaneously be sent to a limiting value.