On the limit set at infinity of a gradient trajectory of a semialgebraic function

Given any C² semialgebraic function f defined on a non-bounded open set of Rn, we prove that the limit of the secants at infinity of a non-bounded trajectory of the gradient of f does exist. As a corollary we find a new sufficient condition to ensure the trivialisation by the gradient flow of f nearby a regular asymptotic critical value at infinity.