In this paper we prove that if 0<p<2 andfis a function analytic in the unit disc δ then, for almost every θ in the set of those θ such thatfhas a non-tangential limit ateiθ. In particular, this holds for almost every θ inRiffis in the Nevanlinna classNand we prove that this result is sharp in a very strong sense. Namely, we prove that if 0<p<2 then given any positive functione(r) defined on[0,1] and tending to 0 asr→ 1 there exists a functionfanalytic in δ and continuous on δ such that, Our results extend to the range 0<p<2 knownresults of zygmundand Hallenbeck and Samotij valid forp= 1.