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**zetafunc.****Guest**

To evaluate

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.

**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

It is because the derivative of the sine function is found through that limit. Evaluating it using something that ot proves is a circular proof.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**zetafunc.****Guest**

Oh of course, I see. So it follows that I can't use the one for cosine either (I think that's used in the proof of derivative of sine). Thanks.

**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

That is right.

You are welcome.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Hi zetafunc;;

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.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,426

hi

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

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**zetafunc.****Guest**

I cannot see on any of those websites an explanation of why they are able to use L'Hopital's rule for sinx/x. They have used it but have not justified their use of it.

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Hi;

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.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,426

Thanks for finding another way to establish the derivative of sin.

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

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,426

He seems to be using x as the angle of the sector and as the distance along the horizontal axis ??

Then a function B jumps out of nowhere ??

Can anybody explain this?

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**zetafunc.****Guest**

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

**zetafunc.****Guest**

And, surely you can just prove that cosine is the derivative of sine by differentiating the definition of sine and showing that it's equal to the definition of cosine...?

**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

zetafunc. wrote:

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.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,426

Does that invalidate the method?

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Hi Bob and Zetafunc;

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.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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