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## #1 2012-10-28 21:49:01

zetafunc.
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### Why can't I use L'Hopital here?

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.

## #2 2012-10-28 22:11:14

anonimnystefy
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### Re: Why can't I use L'Hopital here?

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.

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

## #3 2012-10-28 22:25:22

zetafunc.
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### Re: Why can't I use L'Hopital here?

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.

## #4 2012-10-28 22:55:20

anonimnystefy
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### Re: Why can't I use L'Hopital here?

That is right.

You are welcome.

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

## #5 2012-10-28 23:03:10

bobbym

Online

### Re: Why can't I use L'Hopital here?

In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

## #6 2012-10-28 23:29:09

bob bundy
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### Re: Why can't I use L'Hopital here?

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.

Bob

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

## #7 2012-10-28 23:32:02

zetafunc.
Guest

### Re: Why can't I use L'Hopital here?

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.

## #8 2012-10-28 23:38:55

bobbym

Online

### Re: Why can't I use L'Hopital here?

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.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

## #9 2012-10-29 01:00:44

bob bundy
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### Re: Why can't I use L'Hopital here?

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

## #10 2012-10-29 01:12:41

bob bundy
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### Re: Why can't I use L'Hopital here?

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

## #11 2012-10-29 03:45:02

zetafunc.
Guest

### Re: Why can't I use L'Hopital here?

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

## #12 2012-10-29 03:46:14

zetafunc.
Guest

### Re: Why can't I use L'Hopital here?

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...?

## #13 2012-10-29 03:49:50

anonimnystefy
Real Member

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### Re: Why can't I use L'Hopital here?

#### 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.

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

## #14 2012-10-29 05:40:56

bob bundy
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### Re: Why can't I use L'Hopital here?

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

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

## #15 2012-10-29 11:23:42

bobbym

Online

### Re: Why can't I use L'Hopital here?

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.

http://www.maa.org/pubs/Calc_articles/ma006.pdf

In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.