Does an asymptotically stationary curve in R^n with vanishing acceleration imply vanishing velocity?

For twice continuously differentiable curves $\vec{x}(t) \in \mathbb{R}^n$, does$$\left( \lim_{t \to \infty} \vec{x}(t) = \vec{L} \quad \text{and} \quad \lim_{t \to \infty} \vec{x}''(t) = \vec{0} \right)$$imply$$\lim_{t \to \infty} \vec{x}'(t) = \vec{0}?$$