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

David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx> wrote:
> The way I'd put this is like so: Imagine a solid disk (or other
> convex region) and a closed curve containing it made of rope.
> WLOG one point of the curve is touching the solid. Make that
> point one end of the rope, and fix it to the solid. Pull
> on the other end of the rope - you end up with the rope
> wrapped around the circle (or in the general case wrapped
> around the solid except for some straight segments on the
> rope that correspond to longer parts of the perimeter
> of the inner body) plus some rope left over.

Brilliant! How embarrassingly simple! I am now completely happy. So you
see it was possible to convince me (and our notional 15 year old) of
the limit using only elementary arguments and without resorting to a
technical definition of arc length!

Thank you all so much for your time and patience. The timing couldn't be
better, as I'm about to go to a conference for a few days, and won't get
a chance to read news until I get back...

Only one question remains: why on earth wasn't I taught it like this in
my first course in calculus?!!!

Regards,

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

Relevant Pages

  • Re: sin x / x tends to 1...
    ... >> convex region) and a closed curve containing it made of rope. ... >my first course in calculus?!!! ...
    (sci.math)
  • Re: Depressed LoriB.o.B.
    ... Somehow I can picture a lot of us holding on to that same rope until our ... tippy toes are touching the ground and then we start to "swing" Whee! ... I'll take the extended "rope" & hold on. ...
    (alt.med.fibromyalgia)