Re: how does the concept of a limit work

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.

To derive that sin(x) = x - x^3/6 + x^5/120 - ..., we need to show
that d/dx sin(x) = cos(x) and d/dx cos(x) = -sin(x), both of which
rely on lim_{x->0} sin(x)/x = 1.

Here is a recent thread dealing with lim_{x->0} sin(x)/x = 1.

<http://groups.google.com/groups?threadm=20060306.132118%40whim.org>

Rob Johnson <rob@xxxxxxxxxxxxxx>
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