In an answer elsewhere on this site, @Number points out that L'Hospital's rule can be applied to $\lim_{x\to c} f(x)/g(x)$ whenever $g(x)\to\infty$; no hypothesis on $f$ is needed. That answer gives an application of this "generalized L'Hospital's Rule" in a case where $f$ is unknown.
What is an example of a pair of specific functions $f$ and $g$ for which it is useful to apply the generalized L'Hospital's rule?
Here's my wish list:
It should be reasonably simple to define $f$ and $g$.
We need $\lim_{x\to c} g(x)=\pm\infty$.
$f$ should be unbounded near $c$ (or at least it should not be easy to prove that $f$ is bounded near $c$). Otherwise we can see that $\lim_{x\to c} \frac{f(x)}{g(x)}=0$ without applying L'Hospital's rule.
We want $\lim_{x\to c}f(x)\ne\pm\infty$ (or at least it should not be easy to prove that $\lim_{x\to c}f(x)=\pm\infty$). Otherwise we don't need the generalized version of L'Hospital's rule.
We should be able to evaluate $\lim_{x\to c} \frac{f'(x)}{g'(x)}$.
It should be impossible (or at least not easy) to simplify $\frac{f(x)}{g(x)}$ into a form in which the generalized L'Hospital's rule is not useful.
For example, at $c=\infty$, the functions $f(x)=x\sin x$ and $g(x)=x^2$ satisfy all of these requirements except the last, because $\frac{f(x)}{g(x)}=\frac{\sin x}{x}$, which has a bounded numerator.
