Let $f\colon\mathbb{R}\to\mathbb{R}^n$ be a $C^\infty$ function such that $\lim_{t\to\pm\infty}f^{(k)}(t)$ exists for all $k\in\mathbb{N}\cup\{0\}$. A neat little trick (apparently due to Thomas Hill) shows that $\lim_{t\to\pm\infty}f^{(k)}=0$ for $k\geq 1$. Indeed $$\lim_{t\to\pm\infty}f(t) = \lim_{t\to\pm\infty}\frac{e^t f(t)}{e^t} = \lim_{t\to\pm\infty}\frac{e^t(f(t)+f'(t))}{e^t}=\lim_{t\to\pm\infty}(f(t)+f'(t)),$$ so the claim follows inductively.
Is it possible to draw conclusions about the asymptotic decay of $f'$? For example, is there a $\delta > 0$ such that $\lim_{t\to\pm\infty}f'(t)\cdot e^{\delta |t|}=0$? Are there examples when this is not the case?
