  We study the scaling limit of the rank-one truncation of various beta
ensemble generalizations of classical unitary/orthogonal random matrices: the
circular beta ensemble, the real orthogonal beta ensemble, and the circular
Jacobi beta ensemble. We derive the scaling limit of the normalized
characteristic polynomials and the point process limit of the eigenvalues near
the point 1. We also treat multiplicative rank one perturbations of our models.
Our approach relies on a representation of truncated beta ensembles given by
Killip and Kozhan, together with the random operator framework developed by
Valk\'o and Vir\'ag to study scaling limits of beta ensembles.
