L'H\^opital's rule

(should be done during derivatives, i.e., during 1st quarter)

NB: $0/0$ or $\infty/\infty$ is undefined -algebraically-,
 but it's not undefined -qua limit-: in that case you need to do more
 work, e.g., use L'H\^opital's rule.

NB: in the definition of the derivative, it's -always-
$0/0$ (that's how it works!); one can view L'H\^opital's as a
generalization of the derivative: it compares how quickly two
functions vanish or grow.

NB: L'H\^opital's rule doesn't work for functions which oscillate
faster than they grow, eg, $x+\sin(x^2)$
(can I formalize this?)

mechanical flowchart --------------------

(1)
$0/0$ or $\infty/\infty$:
take der of top and der of bottom
(no quotient rule! it's comparing growth rates, say)

(2)
$0\times \infty$:
rewrite as $\frac{\infty}{1/0}=\frac{\infty}{\infty}$
or $\frac{0}{1/\infty}=\frac{0}{0}$,
then do (1) above.

$1^\infty, 0^0, \infty^0$:
 let $L=\lim ...$, and take $\log$, so you get
 $\log L = \lim \log(...)$;
 now it's $\infty\times 0$ or $0 \times -\infty$
 or $0\times +\infty$; do (2) above
 CAUTION: note that you will find $\log L$;
 you need to take $\exp$ of what you get to find $L$

$\infty - \infty$:
 factor out an $\infty$, and hopefully get
 $\infty \times a$ where $a$ is something obviously bounded
 away from $0$, or $\infty \times 0$; do (2) above.
 (possibly combine fractions over a common base, in applications)

worksheet ---------------------------
(reminder to be careful when dealing with $\infty$)

$\infty - \infty$
This can be $+\infty, -\infty$, or any real number.
EG: $x^2 - x, x-x^2, (x+a)-x$
The last is too easy to be persuasive; here's another way:
$\sqrt{x^2+2ax} - x \to a$
(NB: for $a<0$, the above doesn't make sense for small $x$.
  (students expect $x>0$, but forget that it's going to $\infty$)
 This isn't a problem: it still makes sense for large $x$,
 namely $x>-2a$)

$\infty/\infty$
(similar business: can use $(ax+1)/x$ if wanna make it more
interesting)
