It is easy to make a $C^\infty$ function on $\mathbb{R}^2$ for which the unit tangent vector of a gradient flow converging to an equilibrium has no limit. Think of a function that has a unique critical point, a global minimum at the origin, and such that the gradient of $f$ at the upper half boundary of the circle of radius $r$ makes an angle $\theta(r)$ with the exterior normal, oscillating e.g. within $\pm\pi/4$. There is no difficulty to realize this for a function $C^\infty(\mathbb{R}^2)$ because the norm of the gradient may go to zero extremely fast (I'll add more details at request). Then, all gradient lines will converge to the origin, and all of them will present a oscillating unit tangent vector.