> [!NOTE] Theorem (l'Hopital's rule at infinity) > If $f,g:\mathbb{R}\to \mathbb{R}$ are differentiable and $\lim_{ x \to \infty } f(x) = 0 \quad \text{and} \quad \lim_{ x \to \infty } g(x) = 0 $or $\lim_{ x \to \infty } f(x) = \infty \quad \text{and} \quad \lim_{ x \to \infty } g(x) = \infty $then $\lim_{ x \to \infty } \frac{f(x)}{g(x)} = \lim_{ x \to \infty } \frac{f'(x)}{g'(x)}$provided the second limit exists.