[Haskell-cafe] Re: 0/0 > 1 == False
Kalman Noel
kalman.noel at bluebottle.com
Sat Jan 12 06:23:41 EST 2008
Achim Schneider wrote:
> Actually, lim( 0 ) * lim( inf ) isn't anything but equals one, and
> the anything is defined to one (or, rather, is _one_ anything) to be
> able to use the abstraction. It's a bit like the difference between
> eight pens and a box of pens. If someone knows how to properly
> formalise n = 1, please speak up.
Sorry if I still don't follow at all. Here is how I understand (i. e.
have learnt) lim notation, with n ∈ N, a_n ∈ R. (Excuse my poor
terminology, I have to translate this in my mind from German maths
language ;-). My point of posting this is that I don't see how to
accommodate the lim notation as I know it with your term. The limit of
infinity? What is the limit of infinity, and why should I multiplicate
it with 0? Why should I get 1?
(1) lim a_n = a (where a ∈ R)
(2) lim a_n = ∞
(3) lim a_n = − ∞
(4) lim { x → x0 } f(x) = y (where f is a function into R)
(1) means that the sequence of reals a_n converges towards a.
(2) means that the sequence does not converge, because you can
always find a value that is /larger/ than what you hoped might
be the limit.
(3) means that the sequence does not converge, because you can
always find a value that is /smaller/ than what you hoped might
be the limit.
(4) means that for any sequence of reals (x_n ∈ dom f) converging
towards x0, we have lim f(x_n) = y. For this equation again, we
have the three cases above.
Kalman
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