Re: sin x / x tends to 1...
- From: "NotP" <spam@xxxxxxxx>
- Date: Sat, 03 Sep 2005 15:13:50 GMT
"Darren J Wilkinson" <d.j.wilkinson@xxxxxxxxx> wrote in message
news:dfc836$ou0$1@xxxxxxxxxxxxxxxxxxxx
> 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,
> --
> Dr Darren Wilkinson
> mailto:d.j.wilkinson@xxxxxxxxxxxxxxx
> http://www.staff.ncl.ac.uk/d.j.wilkinson/
It all starts with the power series definition of e^z for complex z.
I realize that requires some hefty work, but the payoff is that the limit
you seek is obvious.
See Rudin, Principles of ..., Chapter 8, or Rudin, Real and Complex
Analysis, Prologue.
.
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- From: Darren J Wilkinson
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