  The goal of this paper is to supplement the large deviation principle of the
Freidlin--Wentzell theory on exit problems for diffusion processes with results
of classical central limit theorem kind. We describe a class of situations
where conditioning on exit through unlikely locations leads to a Gaussian
scaling limit for the exit distribution. Our results are based on Doob's
h-transform and new asymptotic convergence gradient estimates for elliptic
nonlinear equations that allow to reduce the problem to the Levinson case. We
devote a separate section to a rigorous and general discussion of h-transform.
