Single field models of inflation capable to produce primordial black holes usually require a significant departure from the standard, perturbative slow-roll regime. In fact, in many of these scenarios, the size of the slow-roll parameter eta becomes larger than one during a short phase of inflationary evolution. In order to develop an analytical control on these systems, I explore the limit of eta large, and promote 1/eta to a small quantity to be used for perturbative expansions. Formulas simplify, and analytic expressions for the two and three point functions of curvature fluctuations are obtained. I will then discuss the behaviour of loop corrections to inflationary observables in this framework