L'H\^opital's rule:
Here's a few concrete proofs:
Note that you can derive most forms from any other form, by suitable
algebraic manipulations, which shows that they are essentially
equivalent.

The idea behind L'H\^opital's rule is good and simple:
if two functions grow to infinity, then (in the limit) their ratio is
the ratio of their growth rates;
likewise if two functions shrink to zero, then (in the limit) their
ratio is the ratio of their vanishing rate.

$0/0$ at finite point: %------------
this only works if $f, g$ are defined and differentiable at $a$;
the reason for the Cauchy Mean Value theorem is to deal with the
functions possibly being non-differentiable at $a$

suppose we're going to $a^+$
Since the derivative is a linear approximation, then:
$f(a+h)=f(a)+f'(a)h+o(h)$
$g(a+h)=g(a)+g'(a)h+o(h)$
Since $f(a)=g(a)=0$ (assuming we're extending them continuously to $a$),
we get
\begin{multline*}
\frac{f(a+h)}{g(a+h)}
=\frac{f(a)+f'(a)h+o(h)}{g(a)+g'(a)h+o(h)}\\
=\frac{f'(a)h+o(h)}{g'(a)h+o(h)}\\
\end{multline*}
\dots which goes to $f'(a)/g'(a)$

[BTW, if they're not differentiable at $a$,
 you just need to throw in another approximation.]

$\infty/\infty$ at $\infty$: %------------
(I like this one)

we assume $x\to \infty$
(otherwise I think we can use the substitution $y=1/x$ or such)

if $\lim_{x\to \infty} \frac{f(x)}{g(x)}=\frac{\infty}{\infty}$
and $\lim_{x\to \infty} \frac{f'(x)}{g'(x)}=L$
then for $x\gg 0$, we have $\abs{f' -Lg'} < \epsilon$.
Since everything's going off to infinity, we can drop lots of constants:
Recall that:
$f(x)=f(0)+\int_0^x f'$
$g(x)=g(0)+\int_0^x g'$
(note also that since $g\to \infty$, the denominator will be positive
(hence non-zero) for large enough $x$ in the below)

Then
\begin{multline*}
\lim_{x\to \infty} \frac{f(x)}{g(x)}
=\lim \frac{f(0)+\int_0^x f'}{g(0)+\int_0^x g'}\\
=\lim \frac{\int_N^x f'}{\int_N^x g'}
\end{multline*}
where we can drop constants because the top and bottom are going to
infinity (so constants are negligible). By taking $N$ sufficiently
large, this last quantity is arbitrarily close to $L$, hence the limit
is $L$.

------------------------------------------------------------
Formal equivalence of various forms of L'H\^opital's rule:
- finite point and infinite point
These are easily equivalent: just take a change of variables
$y=a+1/(x-a)$
to switch a finite point $a$ and infinity
(verbose form of $y=1/x$)

- $0/0$ and $\infty/\infty$
maxim points out that if you have one of these,
then you can get the other by looking at $\frac{1/g}{1/f}$,
differentiating, obtaining $\frac{g'/g^2}{f'/f^2}$, and cancelling
the $f^2/g^2$ (since you're assuming this goes to a limit!)
(FIXME: is this legitimate? We also want to use the fact that if
$f'/g'$ goes to a limit, then $f/g$ does too -- I think this can work)

------------------------------------------------------------
------------------------------------------------------------

Growth/decay of functions in general
the "correct" way to compare growth rate and decay rate
  is to take $1/y$ (i.e., in codomain)
- to compare decay at a finite point and at infinity,
  take $1/x$ (i.e., in domain)
  (look at stereographic projection (either $\bR$ or $\bC$):
   real circle, complex sphere
   etc. to see that this really switching $0$ and $\infty$ the "right
   way")
(log goes to infinity at same rate in both directions...,
 as shown by $\log 1/x = -\log x$)
(note how -slowly- $\log$ goes to $\infty$ at $0$: every time you half
$x$, $\log$ just goes down by $1$ (or $\log 2$, say).
compare to every time you have $x$, $1/x$ doubles!
