Re: sin x / x tends to 1...

On Sun, 4 Sep 2005 20:28:10 +0000 (UTC), Darren J Wilkinson
<d.j.wilkinson@xxxxxxxxx> wrote:

>N. Silver <mathelp@xxxxxxxxxxxxxxxx> wrote:
>> Darren J Wilkinson wrote:
>> > I know we've been going around the houses a bit, but I think that we may
>> > be getting to the root of my problem. Ignoring the regular/irregular
>> > polygon stuff, which we can regard as mere detail, you (and one or two
>> > other people...) are choosing to _define_ arc length as the sup over
>> > inscribed polygon lengths, knowing that the sin x / x limit immediately
>> > follows. What I am saying is that this is essentially the same as
>> > _assuming_ the sin x / x limit,
>>
>> Yes
>>
>> > because you _believe_ it to be true.
>>
>> No, it can be proved using series but not so
>> easily from first principles..
>
>But the series doesn't exist without the limit. You can start by
>_defining_ sin with a series, but you still need to convince me that it
>corresponds to the usual geometric definition without any slight-of-hand.

This is getting to be a little, um, ...

I showed you how to start with the power series definition and
show that it corresponds to the usual geometric definition of sin.
Your reply was that the fact that the length of a differntiable
curve is equal to that integral was motivated by a desire to
show that sin(x)/x -> 1, so it didn't count. This made no
sense to me at first - made a little more sense after you
explained. So explained carefully exactly how it all followed
from the _standard_ definition of arc length.

Until today I assumed this was all a good-faith attempt to
resolve honest onfusion on your part. But I'm beginning to
think that you're just trolling. Here's a few facts:

(i) The definition I gave for the length of a curve is
absolutely standard - comes up all the time in many
contexts, most of which have nothing to do with sin(x)/x.
For example, it's the standard definition for the length
of a curve in a general metric space.

(ii) The fact that sin(x)/x -> follows immediately from
the definition of the length of a curve (how immediately
depending on how many of the purely technical details
we write down.)

You conclude from (ii) that the definition was chosen
so as to allow us to prove that sin(x)/x -> 1. So
things are still circular. But you don't seem to be
allowing even the possibility that you're simply
wrong about something: It could be that the definition
of the length of a curve is standard because it's
a perfectly reasonable, simple, and intuitively
plausible definition; then it could be that (i)
and (ii) don't indicate some vast conspiracy on
the part of definers of arc length, it could be
that the fact that sin(x)/x -> 1 is simply not
as mysterious as you think it is!

I mean, look. Before we can prove anything about
anything we need definitions of the terms involved.
I begin to get the idea that no matter what anyone
says about this your reaction is going to be that
since the limit follows from those definitions it
follows that the definitions were chosen just for
this purpose, so the argument is circular.

I find it hard to believe that that's going to
be your _honest_ reaction no matter what anyone
says, but it seems like that's going to be what
you _say_. Hence the conjecture about "trolling".

So answer me a question. Exactly how would someone
with no hidden agenda re sin(x)/x _define_ the
length of a curve?

>Regards,


************************

David C. Ullrich
.

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