I discovered while teaching Calc 2 that if you apply L'Hopital's rule to $\frac{x}{\sqrt{x^2+1}}$ you get $\frac{\sqrt{x^2+1}}{x}$, and if you apply L'Hopital again you get $\frac{x}{\sqrt{x^2+1}}$ back. In other words the L'Hopital operator has a cycle of order two.
EDIT (Thanks KennyTM): "I suppose the L'Hopital operator should be defined on equivalence classes of pairs $(f(x),g(x)$ of differentiable functions with the fractional equivalence: $(f(x),g(x))\equiv(h(x),k(x))$ if and only if $fk=gh$." This does not work. But it doesn't really take pairs of functions to pairs of function, either. So the first problem is to find out how it is an operator.
Has anyone ever studied this operator? Wikipedia tells me nothing.
Yes, I know it is easier to find the limit by dividing through by $x$, but some students want to apply L'Hopital to everything.