Re: how does the concept of a limit work

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?

Rob Johnson <rob@xxxxxxxxxxxxxx>
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