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


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


Sin(x)/x = 1 - x^2/3! + ...
Cos(x) = 1 - x^2/2! + ...

As long as x > 0, cosx =/= sinx. So is it correct to say that sinx/x
approaches cosx as x appropaches zero? I say no!

Regards

Bill J.

.

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