Using tools from Tropical and Non-Archimedean Geometry, we show that there is a tight relationship between the following two concepts of real inflection of real linear series defined on real algebraic curves: 1. that of complete series on hyper-elliptic curves, and 2. that of incomplete series on elliptic curves. Concretely, the case (1) can be degenerated to the case (2), and the case (2) can be regenerated to the case (1). This interplay gives us two products: 1. A limit linear series on a (marked) metrized complex of (real) algebraic curves. By this we mean a marked tropical curve with real models. 2. A 2-dimensional family of polynomials generalizing the division polynomials (which are used to compute the torsion points of elliptic curves) This is a joint work with I. Biswas (TATA, India) and E. Cotterill (UFF, Brazil).