Re: abundance of irrationals!)

For example, the (correct) statement
> > >> the limit of f(x) = x sin( 1/x ) as x tends to 0 is 0.


> > >> may be expressed as a cumbersome but precise statement about the
> > >> behaviour of the function f near zero. The words "end" or "stop"
> > >> never occur in the translated statement.
> >
> > >near zero, or at zero? Most limits give pretty precise values for what is
> > >happening at that precise point.
> >
> > Oh? What is "happening at [the] precise point" x=0 for this function f ?
> > Are you saying the limit at x=0 does not exist because f(0) is undefined?
> Well f(0) is undefined, why? Because sin(1/0), or sin(oo), is undefined. But,
> we know that sin is always between 1 and -1 and so x sin(1/x) at x=0 is
> equivalent to 0*(some number between 1 and -1), which is always 0.

What do we see as THE function here. I thought it was f(x) = x . sin (1/x). It's meant as function.

You talk about how to compute it. But that is another question. And in mathematics we canll such a limit that exist and also nice analytical is an analytical extension. But always we have to look at the whole function. Why do you write it down then?

That means for the whole function that limit disappears. We have just extended the image and can add 0 to our domain. It's a standard way in mathematics.

Look at the gamma function as an example. On the naturals its's the faculty, and on the whole Complex plain it's a complex function, only the 0, -1,-2 etcetera are real poles. Not extendable in the poles. Infinite is not a part of the normal complex C.

I hope not again the same discussion and now about C.

Ed
.

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