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

David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx> wrote:
> First I should say that it seems to me that the simplest
> thing really is to define sin(x) by the power series. Then
> the proof that sin(x)/x -> 1 is very simple (it's not clear
> to me whether you said in a post below that this was not
> clear to you), and it's not too hard to show that this
> turns out to be the same as a geometric definition.
>
> But if you say that's not elementary enough fine. Let's
> talk about sin defined "geometrically", and let's ignore
> the fact that just _defining_ what the area of a region
> _is_ takes some not-so-elementary work; let's assume,
> as in a typical elementary context, that we know what
> areas are.

OK - I guess I should have been more explicit - by "elementary" and
"nice", I was meaning the kind of explanation that could be given to a
bright 15 year old. So, techniques based on power series, integrals,
differential equations, etc., are really not what I had in mind. The
kind of argument you gave was exactly the kind of thing I was after.
Now, if I understood it correctly, you showed that sin x / x tended to
one when x was measured in "aradians". Is there an easy way to show that
aradians and radians are the same?

Regards,

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

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