Given a family of projective varieties X_t over a complex disk with one singular fibre over 0, there is a ‘limit Hodge structure’ on the cohomology of X_t as t approaches 0. For certain families of Calabi-Yau 3-folds, we can use mirror symmetry to compute the limit structure precisely. Assuming standard conjectures from the theory of mixed motives, this is used to explain the occurrence of zeta(3) in the constant term of the prepotential on the Kaehler moduli space. This circle of ideas leads also to insight on a conjecture of Candelas, de la Ossa, and van Straten on limit Frobenius structures in p-adic mirror symmetry. This is report on joint work with Alex Cruz, Michael Harris, and Wenzhe Yang.
