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 $\frac{\sqrt{x^{2n}+c}}{x^n}$.

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.
