Re: opponents of taylor and l'hospital ?

In article <10322212.1210025782327.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
amy666 <tommy1729@xxxxxxxxxxx> wrote:
In article <4VseSEovwEPDM9afsm8KhiLuj141@xxxxxxx>,
"[Mr.] Lynn Kurtz" <kurtz@xxxxxxxxxxxxxxx> wrote:

On Mon, 05 May 2008 00:27:58 GMT, Gerry Myerson
<gerry@xxxxxxxxxxxxxxxxxxxxxxxxx> wrote:


There's an objection to using l'Hopital to do
limit as x goes to zero of (sin x) over x.

The objection is that to use l'Hopital, you need
to know
the derivative of sin x --- but to know the
derivative
of sin x, you have to know the limit as x goes to
zero

why do you need the limit as x goes to zero to compute the derivatative of sin x ??

its not that hard ?

1) its very well known

Proof by common knowledge. Right below proof by intimidation.

2) you can use taylor series

How does one develop the taylor series for sine without knowing how
to differentiate sine?

3) you can use nilpotent numbers h =/= 0 and h ^ 2 = 0

You lost me there. sine is defined on the real line and can be
extended to the complex plane, but neither of these fields have
zero divisors.

of (sin x) over x, so using l'H is circular (no
pun intended)
reasoning.

I don't recall ever seeing the limit for sin(x)/x
developed in any
calculus book by using L'Hospital's rule. Have you?

well actually i have.

dont remember which one , and it might not have been a good one , but im sure i have.

It would be nice to see a reference. They are much more convincing.


No, not in any book - but on student papers, yes,
many times.
The objection is really to assigning that limit as a
problem
when you're teaching l'H.

why ? you can use that example to demonstrate l'H.

Probably because of the circular nature of the reasoning.

I have nothing against using Taylor series or L'Hospital's rule,
except when used circularly in a proof. If one has already derived
the derivative of sine, then there is nothing wrong with using that
result to rederive the limit of sin(x)/x. However, since this limit
was already derived, why rederive it?

I believe that the thread that started this was "Real Analysis Help
... Sequences" when William Elliot said, "lim(x->0) (sin x)/x = 1.
That is calculus." and I replied, "That is circular, at least if
you are thinking of using L'Hospital." I would have kept quiet if
he had said, "That is pre-calculus." Pre-calculus generally covers
trigonometry and limits; definitely enough to handle this limit.
Pre-calculus also precludes the use of L'Hospital.

Rob Johnson <rob@xxxxxxxxxxxxxx>
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