On Hessian Limit Directions along Gradient Trajectories Given a non-oscillating gradient trajectory $|\gamma|$ of a real analytic function $f$, we show that the limit $\nu$ of the secants at the limit point $\mathbf{0}$ of $|\gamma|$ along the trajectory $|\gamma|$ is an eigen-vector of the limit of the direction of the Hessian matrix $\operatorname{Hess} (f)$ at $\mathbf{0}$ along $|\gamma|$. The same holds true at infinity if the function is globally sub-analytic. We also deduce some interesting estimates along the trajectory. Away from the ends of the ambient space, this property is of metric nature and still holds in a general Riemannian analytic setting. Keywords:gradient trajectories, non-oscillation, limit of Hessian directions, limit of secants, trajectories at infinityCategories:34A26, 34C08, 32Bxx, 32Sxx