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

In article <l2arh1hecg8ugll8o3fpdb1eih9tk3lju9@xxxxxxx>, David C.
Ullrich <ullrich@xxxxxxxxxxxxxxxx> wrote:

> On Tue, 06 Sep 2005 05:51:51 -0700, Ronald Bruck <bruck@xxxxxxxxxxxx>
> wrote:
>
> >In article <dfjj1q$c0r$1@xxxxxxxxxxxxxxxxx>, Darren J Wilkinson
> ><d.j.wilkinson@xxxxxxxxx> wrote:
> >
> >> Ronald Bruck <bruck@xxxxxxxxxxxx> wrote:
> >> > In article <dfi87o$gl1$1@xxxxxxxxxxxxxxxxx>, Darren J Wilkinson
> >> > > Now let's look at circumference. We can look at the perimeter of the
> >> > > inscribed N-gon, and see what happens as N increases. There isn't much
> >> > > point looking at the perimeter of the circumscribed polygon, because it
> >> > > isn't "obvious" that its perimeter is greater than the circumference of
> >> > > the circle.
> >> >
> >> > Sure it is. Let L_n be the perimeter of the inscribed polygon, M_n the
> >> > perimeter of the circumscribed polygon (both regular, of n sides). A
> >> > little work shows L_n is increasing and M_n is decreasing (but we
> >> > really only need this for powers-of-2, which is still less work).
> >> > Manifestly L_n < M_n, because the circumscribed polygon is a scalar
> >> > multiple > 1 of the inscribed polygon, rotated slightly--which clearly
> >> > doesn't affect the perimeter. We conclude that L_n < M_m for any m and
> >> > n (because L_n < L_{mn} < M_{mn} < M_m).
> >> >
> >> > If the perimeter of the circle is defined as the sup of the L_n, it's
> >> > now clear that M_n is > the perimeter.
> >>
> >> Doh! That is _assuming_ the (modern) definition of arc length.
> >
> >Arc length for POLYGONS is trivial.
>
> Huh? You're using the definition of arc length for a CIRCLE.

That's only a remark. My POINT is that the inscribed polygon has
shorter arc length than the circumscribed polygon. That's all you need
to prove the desired limit. I'm more surprised nobody has challenged
my assertion that L_n is increasing. (But, as I say, all you really
need is for L_{2^n} to be increasing, and that's trivial.)

The fact that L_n < M_m means the arclength of the circle is <= M_m.
The point of the remark is to explain WHY the arclength of the circle
is less than the circumscribed perimeter (which, while intuitive, is
otherwise difficult to prove).

As for my assuming the "modern" definition of arclength: there doesn't
seem to BE any other definition; the Greeks took arclength as a
primitive concept, and didn't really have a definition.
.

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