Re: how does the concept of a limit work

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In article <20060423.013538@xxxxxxxx>, Rob Johnson <rob@xxxxxxxxxxxxxx>
wrote:

In article <rTs2g.8$Rx4.7@xxxxxxxx>,
"Stephen J. Herschkorn" <sjherschko@xxxxxxxxxxxx> wrote:
Rob Johnson wrote:

In article <4aur9aFuuubvU1@xxxxxxxxxxxxxx>,
Jose Carlos Santos <jcsantos@xxxxxxxx> wrote:


Colleyville Alan wrote:



i am still confused about the findamental limit.
can it be proven that as x->0 : sinx/x -> 0


No. Rather, it can be proven that, as x -> 0, sin(x)/x -> 1.


Of course, the *way* of proving it depends upon the definition of sine
that a person is working with.


Could you elaborate on that comment?


Sure. If you define sin(x) as x - x^3/3! + x^5/5! - x^7/7! + ..., then
it is rather easy to prove the statement. If you define it in
[-pi/2,pi/2] as the inverse of the arcsine function, which in turn is
defined as the integral of 1/sqrt(1 - t^2) where _t_ goes from 0 to _x_,
then it's a bit harder.



How about if we define the sine of an angle as the ratio of the
opposite side divided by the hypotenuse in a right triangle?
To me, that is THE definition of sine. Everything else is derived
from that.


That is fine for an elementary introduction to the topic. However, the
approach is not rigourous until you define angle and its measure.
Starting with arcsine is an basically the usual way to do this.

I thought that angle was defined as the circular arclength divided by
the radius, which can easily be shown to be twice the area of a wedge
over the sqaure of the radius. I can see that these lead to the arc
sine integral, but I don't think that the arcsine integral is needed
to define the angle. In the proof of the limit I cited, the angle is
measured as twice the area of a circular wedge in a unit circle.

Am I being too naive?

It's unclear. The problem with elementary geometry is that it's MUCH
harder to axiomatize and prove things carefully than you would imagine
from high-school textbooks. I know Hilbert did this, but I'm not very
familiar with his work, and in particular I don't know how arclength
and angle are handled in a careful treatment.

Better-known today is Birkhoff's axiomatization, but I don't see
anything in that about angle measure. I did find a reference to
MacLane's axioms, at

<http://libraryofmath.com/math/Foundations_of_Geometry/The_MacLane_Postu
lates/The_MacLane_Postulates.html>

(presumably this is the same Saunders MacLane whom I took courses from)
and MacLane does have axioms for angular measure. But it looks like it
would be a lot of hard work to build up to measuring angles in a
circle, and connecting that to arclength.

What we need is somebody who's an expert in the foundations of geometry
to weigh in on this. By that I mean somebody who KNOWS, not just
somebody who has an OPINION.

MY opinion is that it's so hard that most folks just punt and use the
series definition; and most of these alternatives (like inverse of
arcsin, which I've developed in other threads) are motivated by the
fact that students haven't had power series yet.

It does seem paradoxical that the most intuitive treatment might be the
most difficult.

--
Ron Bruck
.

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