The statement of l'Hopital's rule found in Rudin's Principles of Mathematical Analysis (page 109) is:
$5.13$ $\,\,$ Theorem $\,\,\,\,\,\,\,$ Suppose $f$ and $g$ are real and differentiable in $(a,b)$, and $g'(x)\neq 0$ for all $x\in (a,b)$, where $-\infty\leq a < b\leq +\infty$. Suppose
$$
\frac{f'(x)}{g'(x)}\to A\,\textrm{ as }\,x\to a.
$$
If
$f(x)\to 0$ and $g(x)\to 0$ as $x\to a$, or if $g(x)\to +\infty$ as $x\to a$, then
$$
\frac{f(x)}{g(x)}\to A\,\textrm{ as }\,x\to a.
$$
The analogous statement is of course also true if $x\to b$, or if $g(x)\to-\infty$ [...]. Let us note that we now use the limit concept in the extended sense of Definition $4.33$.
It seems that what $A$ is supposed to be is a little ambiguous. Real? Possibly infinite? However, we can resolve this enigma with a quick look at definition $4.33$:
$4.33$ $\,\,$ Definition $\,\,\,\,\,\,\,$ Let $f$ be a real function defined on $E\subset \Bbb{R}$. We say that
$$
f(t)\to A\,\textrm{ as }\,t\to x,
$$
where $A$ and $x$ are in the extended real number system, if for every neighborhood $U$ of $A$ there is a neighborhood $V$ of $x$ such that $V\cap E$ is not empty, and such that $f(t)\in U$ for all $t\in V\cap E$, $t\neq x$.
So indeed, $A$ is allowed to be infinite (it's in the extended real number system)! We also see that Rudin treats the cases of $A = \pm\infty$ in the proof of l'Hopital's rule, so if the limit of the quotient of derivatives is infinite, the original limit must have been also. That is, there is no example (let alone a simple one) where the limit of the quotient of derivatives is infinite, but the original limit is finite.
A precondition in l'Hospital's Rule is that in order for it to apply, the limit
$$
\lim_{x\to\infty}\frac{f'(x)}{g'(x)}
$$
must exist (but is allowed to be $\pm \infty$). In this case, the limit does not exist, so it does not apply.
Best Answer
This is far from rigorous, but the way I like to think about L'Hospital's Rule is this:
If I have a fraction whose numerator and denominator are both going to, say, infinity, then I can't say much about the limit of the fraction. The limit could be anything.
It's possible, though, that the numerator goes slowly to infinity and the denominator goes quickly to infinity. That would be good information to know, because then I would know that the denominator's behavior is the one that really swings the limit of the fraction overall.
So, how can I get information about the rate of change of a function? This is precisely the kind of thing a derivative can tell you. Thus, instead of comparing the numerator and denominator directly, I can compare the rate of change (i.e. the derivative) of the numerator to the rate of change (i.e. the derivative) of the denominator to determine the limit of the fraction overall. This is L'Hospital's Rule.