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

David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx> wrote:
> On Sun, 4 Sep 2005 20:09:00 +0000 (UTC), Darren J Wilkinson
> >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.
>
> Yes, the fact that sin(x)/x -> 1 follows immediately. But that's
> not the reason "I" "chose" to define arc length that way. That
> _is_ the _standard_ definition of arc length!
>
> How _should_ arc length be defined, in your opinion?

You mis-understand me. I'm perfectly happy with the definition of arc
length you adopt. But if you think about what it is saying, it is saying
that as you make your "ruler" smaller, the discrepancy between "true"
arc length and the length of the "ruler" will become insignificant. It is
not something you are proving - it is something that you are (quite
reasonably) asserting. And if you think about what the sin(x)/x limit is
saying, it is saying exactly the same thing in the special case of a
circle. So all I'm saying is that it is a bit ingenuous to talk of
"proving" the sin(x)/x limit when your definition of arc length is
assuming something much stronger.

> You're
> saying I chose that definition because it would give the
> desired limit - how you can _think_ you know this about
> why I did something I can't imagine, but you're wrong
> about that. I didn't "choose" that definition at all,
> it's the definition I _learned_.)

As I've said, I think you mis-interpreted my (admittedly slightly
provocative) choice of words.

> Two questions for you: How _should_ I define arc length,
> and how, in your opinion, would I have defined it if
> I hadn't been determined to justify that limit?

So, as I said above, I'm happy with your definition of arc length. But
hypothetically speaking, if the sin(x)/x limit "turned out to be" (say)
2/3 (I AM NOT SUGGESTING FOR A SECOND THAT I THINK THIS!), then you
might (say) define arc length to be 3/2 times your definition, for
example... I don't think it is going to be productive to persue this
further, however! ;-)

Regards,
--
Dr Darren Wilkinson
mailto:d.j.wilkinson@xxxxxxxxxxxxxxx
http://www.staff.ncl.ac.uk/d.j.wilkinson/
.

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