Coaxioms have been recently introduced to enhance the expressive power of inference systems, by supporting interpretations which are neither purely inductive, nor coinductive. This paper proposes a novel approach based on coaxioms to capture divergence in semantic definitions by allowing inductive and coinductive semantic rules to be merged together for defining a unique semantic judgment. In particular, coinduction is used to derive a special result which models divergence. In this way, divergent, terminating, and stuck computations can be properly distinguished even in semantic definitions where this is typically difficult, as in big-step style. We show how the proposed approach can be applied to several languages; in particular, we first illustrate it on the paradigmatic example of the λ-calculus, then show how it can be adopted for defining the big-step semantics of a simple imperative Java-like language. We provide proof techniques to show classical results, including equivalence with small-step semantics, and type soundness for typed versions of both languages.