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

David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx> wrote:
> On Sun, 4 Sep 2005 08:57:40 +0000 (UTC), Darren J Wilkinson
> >Anyway, I think this thread has now served its purpose. Many thanks to
> >everyone who posted. I'm much clearer now what the issues are. I think
> >if I really wanted to resolve this to my own satisfaction, then I
> >probably could, probably starting with a power series definition. But it
> >certainly wouldn't be "nice"...
>
> Actually it is nice, so there.

;-)

> >My conclusion as far as our notional bright 15 year old is concerned, is
> >to first give an argument for the area of a circle which doesn't (appear
> >to) depend on the sin x / x limit, and then use the usual area argument
> >given in standard texts. My problem with that is that its very hard to
> >do. Even the "derivative of area is circumference" argument I feel
> >relies implicitly on the sin x / x limit,
>
> I don't see how. The idea is that you consider the region between
> a circle of radius r and a circle of radius r + dt, you argue that
> that's equal to the circumference times dt because it's just a
> strip of width dt and length = circummference. (Of course that's
> not a _proof_...)

So my thinking here was that the strip isn't actually rectangular, and
if you wanted to convince yourself that it had the same area as the
corresponding rectangular strip, you might bump into the limit... But
either way, as you say, it isn't really a proof.

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

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