Circular uses of L’Hôpital’s rule
If you try to find the $\lim_{x\to \infty}\frac{\sqrt{x^2+1}}{x}$ using L’Hôpital’s rule, you’ll find that it flip-flops back and forth between $\frac{\sqrt{x^2+1}}{x}$ and $\frac{x}{\sqrt{x^2+1}}$.
Are there other expressions that do a similar thing when L’Hôpital’s rule is applied to them? I already know that this applies to any fraction of the form $\frac{\sqrt{x^{2n}+c}}{x^n}$.
From $$\frac{f'(x)}{g'(x)}=\frac{g(x)}{f(x)}$$ we get $$2f(x)f'(x)=2g(x)g'(x)$$ and integrating $$f(x)^2=g(x)^2+C$$ or $$f(x)=\pm\sqrt{g(x)^2+C}$$ provided that $C\ge-g(x)^2$ for every $x$ in the domain of $g$.
Here is a list of limits related to the OP question. L'Hospital's rule cannot be used for each.
$\lim\limits_{x\rightarrow 0}\dfrac{x^{2}\sin \left( \frac{1}{x}% \right) }{\sin x}$
$\lim\limits_{x\rightarrow \infty }\dfrac{x-\sin x}{x+\sin x}$
$\lim\limits_{x\rightarrow \infty }\dfrac{\left( 2x+\sin 2x\right) }{% \left( 2x\sin x\right) e^{\sin x}}$
$\lim\limits_{x\rightarrow \infty }\dfrac{\sqrt{2+x^{2}}}{x}$
$\lim\limits_{x\rightarrow 0}\dfrac{1-\cos x}{\cos x}$
$\lim\limits_{x\rightarrow \infty }\dfrac{\sqrt{9x+1}}{\sqrt{x+1}}$
$\lim\limits_{x\rightarrow 0}\dfrac{\sqrt{x}}{\sqrt{\sin x}}$
$\lim\limits_{x\rightarrow 0}\dfrac{\cot x}{\csc x}$
$\lim\limits_{x\rightarrow \left( \pi /2\right) ^{-}}\dfrac{\sec x}{% \tan x}.$
You're basically asking if we can find functions $f, g$ such that:
$$\frac{f'}{g'} = \frac{g}{f}$$ i.e.
such that $$ff'=gg'$$
And in this particular case, you have $ff'=x$, of which the solutions are of the form $x \mapsto \sqrt{x^2 + c}$ (you can see that $x \mapsto x$, $x>0$ is a particular case when $c=0$).
Then you have found the solutions for $ff' = nx^{2n-1}$.
Now for any function $v$, you can look for solutions of the equation:
$$ff' = v$$ and if you find two solutions $f, g$ that don't touch $0$, the same phenomenon will occur with their ratio.
EDIT: We could probably look for bigger cycles, like
$$\frac{f'}{g'}=\frac{u}{v} \text{, and } \frac{u'}{v'} = \frac{g}{f}$$
But it looks a little bit more difficult to study, as it feels like a lot of things can happen.