Can L'Hôpital's rule be applied to the limit inferior of a sequence?

Question

Let $f$ and $g$ be continuous on an interval $I$ and differentiable on $I\setminus\{a\}$ for some $a\in I$, such that $f(a)g(a)$ is of an indeterminate form. Does $$\liminf_{x\rightarrow a}\frac{f'(a)}{\frac{d}{dx}\left(g(a)^{-1}\right)}=\liminf_{x\rightarrow a}\frac{f(a)}{g(a)^{-1}},$$ hold? If so, how can this be proven as a specific case of L'Hôpital's rule? The question is motivated by analysing oscillatory limits, specifically that of the distance function squared $$\lVert x\rVert^2=\frac{\arccos(\cos(2\pi x))^2}{4\pi^2},$$ which of course has a limit inferior of $0$. Any help would be much appreciated.