Extensão: Razão entre circulos

You may wonder

Gee, L'Hopital's rule is used to calculate limits, which is one of the most basic concepts in Calculus. Why is it in the Frontiers of Calculus??

Well, let's investigate a cute example of how to use L'Hopital's rule. Remember when we computed the derivative of $\sin (x),$ we needed to evaluate $$\lim_{x\to 0} \frac{\sin(x)}{x}$$

If you need a refresher, click here.

Ok, so to evaluate this limit, we just use L'Hopital's rule, right?

So we just used L'Hopital's rule to get $$\lim_{x\to 0} \frac{\sin(x)}{x} = 1$$

Hmm this sounds an awful lot like cheating. We just used L'Hopital's rule to compute a limit from a long time ago. Why couldn't we have just learned L'Hopital's rule earlier??

So to use L'Hopitals rule, we needed the derivative of $\sin(x),$ but to compute $\sin(x)$ using the formal definition of derivative, we needed to know $\lim_{x\to 0} \frac{\sin(x)}{x} = 1$ to define the derivative of $\sin$!!

Hmmm, how about,

We teach L'Hopital's rule after derivatives because using L'Hopital's rule requires derivatives to be defined.

And in the $\sin(x)/x$ example, we how we need to be careful using L'Hopital's rule to reprove earlier results :).

(Note: L'Hopital's rule is true, so this explanation does not caution against using L'Hopital's rule in general, but rather cautions against using it specifically in developing theory, since then we need to be careful about which stuff implies which other stuff)