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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
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón
#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
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón
#5 2012-10-28 23:03:10
- bobbym
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Re: Why can't I use L'Hopital here?
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& … vkuusUrAkg
and (cos(x))/x here:
https://docs.google.com/viewer?a=v& … MI6SjOYjvA
Also, it does get the right answer.
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#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.
https://docs.google.com/viewer?a=v& … vkuusUrAkg
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.
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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
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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.
http://www.youtube.com/v/PNTnmH6jsRI
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#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
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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
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón
#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
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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.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.