The tangent cone of a curve contains all the possible tangents of the curve at some fixed point. The differentiability problem of geodesics in sub-Riemannian geometry can thus be considered via the study of tangent cones. The dilation structure of Carnot groups is particularly convenient for this study, since the tangents of geodesics become themselves (infinite) geodesics in the same Carnot group. For this reason also the asymptotic properties of infinite geodesics are of interest, since such properties will give restrictions on the possible tangents of geodesics. In this talk I will discuss our recent results that every tangent of a geodesic is also a geodesic in a Carnot group of lower step and that every asymptotic cone of a geodesic is also a geodesic in a Carnot subgroup of lower rank. This is joint work with Enrico Le Donne.