Re: abundance of irrationals!)

In article <MPG.1ce2ec19ab7de659989bb6@xxxxxxxxxxxxxxxxxxxxxxxxx> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> writes:
> Dave Rusin said:
....
> > Informally, perhaps. I'm sure many mathematicians would say they've
> > reached their limit with this discussion. But when they're using the
> > word "limit" in a mathematical sentence, they have a precise meaning
> > for it, which does not involve the terms "end" or "stopping point"
> > at all. 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.

Near zero, never at. The function above is not defined for x = 0, so in
the definition of limit f(0) can not even be used. And it is not about
what is happening at that point, but what is happening near that point.
That that limit is 0 merely means (informally) that the function value
comes closer and closer to 0 when we approach 0, and that we can get as
close as we wish by approaching 0 far enough.
--
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