This is how I am looking at it.

It all depends on how the proof of the d(sin x) is presented in the textbook that also proves your limit later on. If they use the proof using first principles then proving the limit is then circular. But if they prove D(sin(x)) in another way then using L' Hospitals rule on your limit is fine.

Hi Bob;

Here is a geometric proof that D(sin(x))= cos(x). If you use it instead of the one that uses first principles ( uses the limit in question ) then there is no circularity and using L' Hospitals later on in the book is fine.

I could not find fault with the geometric proof, here it is. You can check it too.

]]>As long as you can prove the format without recourse to circular arguments.

But can you? Most explanations seem to rely on differentiation.

Bob

ps. I've been looking at

http://math.stackexchange.com/questions … x-circular

on and off all afternoon, and I think I've found three flaws in the proof.

Bob

]]>Why can't you just use the definition of sine and differentiate that?

Thatis not the definition of sine. That is a result derivable from Euler's theorem.

]]>Then a function B jumps out of nowhere ??

Can anybody explain this?

Bob

]]>There are several lines in that proof that are obscure to me at the moment ... but I'll work on it.

Bob

]]>bob bundy wrote:

Just because lots of internet sites use it, does not make it valid.

Nor does getting the right answer.

This document does identify the argument as circular in example 9.

Not just sites but published PDF's, MIT lecture on youtube, and textbooks!

Elementary Calculus by Keisler ex 21 section 5.2

If many sites and books can not convince anyone of its validity why does only one site prove it invalid? Isn't it possible the offhand suggestion of circularity might be wrong? I am just saying it is not even clear among mathematicians and textbooks.

You can establish the derivative without resorting to that limit so it is not necessarily circular.

http://math.stackexchange.com/questions … x-circular

Here it is used in the MIT lecture on youtube.

3rd example.

I agree it is circular if and only if the only proof of the derivative of sin is by using this result. Anyone know another proof?

Just because lots of internet sites use it, does not make it valid.

Nor does getting the right answer.

This document does identify the argument as circular in example 9.

https://docs.google.com/viewer?a=v&q=ca … vkuusUrAkg

Bob

]]>I think your friend is incorrect in stating this is circular.

It is used here:

http://www.analyzemath.com/calculus/lim … _rule.html

http://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule

http://mathinsight.org/lhospitals_rule_refresher

http://www.math.hmc.edu/calculus/tutorials/lhopital/

http://www.enotes.com/math/q-and-a/use- … let-299844

https://docs.google.com/viewer?a=v&q=ca … vkuusUrAkg

and (cos(x))/x here:

https://docs.google.com/viewer?a=v&q=ca … MI6SjOYjvA

Also, it does get the right answer.

]]>You are welcome.

]]>why can't I just use L'Hopital's rule? I was told that I couldn't because it's a circular proof, but I don't understand why... I know you evaluate this using the squeeze theorem (to show that it's 1) but L'Hopital's rule seems a lot easier to use here.

]]>