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

On Sat, 03 Sep 2005 09:29:23 -0700, quasi <quasi@xxxxxxxx> wrote:

>On Sat, 3 Sep 2005 13:22:46 +0000 (UTC), Darren J Wilkinson
><d.j.wilkinson@xxxxxxxxx> wrote:
>
>>I've a question about the limit of sin x / x as x tends to zero. Of
>>course, it's 1 (I think), but I've never seen a satisfactory proof. The
>>proof I was given, and the proofs I can find in standard texts all rely
>>on knowing the area of a circular sector. However, to know the area of a
>>circular sector, one must know the area of a circle. All the derivations
>>I know for the area of a circle make use (either directly or indirectly)
>>on the sin x / x limit, and there lies my disatisfaction. Of course it's
>>easy to get the upper bound of one, and I'm happy to use the area
>>argument to establish the existance of a limit. However, it seems to be
>>surprisingly awkward to establish the obvious lower bounds (such as cos
>>x) using elementary arguments. Does anyone know a nice proof?
>>
>>Regards,
>
>power series

hmm, well ... maybe this also uses that limit (if you start with
sin(x) defined geometrically).

But if you _define_ sin(x) as a power series, then you can prove the
limit and also the geometry.

quasi
.

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