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 $g$.
