Re: Limit[f[x], x->a] vs. f[a]. When are they equal?
- To: mathgroup at smc.vnet.net
- Subject: [mg118388] Re: Limit[f[x], x->a] vs. f[a]. When are they equal?
- From: Richard Fateman <fateman at cs.berkeley.edu>
- Date: Tue, 26 Apr 2011 04:43:27 -0400 (EDT)
- References: <ip3lub$r9a$1@smc.vnet.net>
On 4/25/2011 4:29 AM, Andrzej Kozlowski wrote:
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> I forgot to add that the function Sin is certainly not continuous at infinity (Sin[1/z]
has an essential singularity at 0)
so there is no reason why it's value at there should agree with its limit.
In fact, it definitely should not do so.
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> Andrzej Kozlowski
>
Hm. continuity (excerpt from wikipedia, but whatever)
The limit of f(x) as x approaches c through domain of f does exist and
is equal to f(c); in mathematical notation, \lim_{x \to c}{f(x)} = f(c).
If the point c in the domain of f is not a limit point of the domain,
then this condition is vacuously true, since x cannot approach c through
values not equal c. Thus, for example, every function whose domain is
the set of all integers is continuous.
........
There is now a kind of semantic gap, it seems to me. There are, perhaps,
several gaps.
One gap is between the mathematical concept of limit and the result of
computing Limit[] in Mathematica. Often they corresponds. Sometimes
they differ. I see this difference appearing when the result in
Mathematica is an Interval, and sometimes when the result is
ComplexInfinity or its relatives. I do not believe that the
mathematical result 1/2+ i*(infinity) is the same as Mathematica's
I*Infinity or ComplexInfinity. See below for a proof.
Another gap is around special symbolic values that might be used as
limit points but which have no neighborhoods. Look at the definition
above. Is there some set of Mathematica values x1, x2, x3, ...., xn, ...
all not equal to c, such that their distance from c becomes smaller,
when c = Infinity? Apparently not, since Abs[x-Infinity] is Infinity for
all x.
So we fall into the vacuously true clause, in which f[c] is the limit.
This is, I think, a consequence of introducing oddball pseudo-real
objects into your computer system (notions like indeterminate,
intervals, infinity).
It's possible I've overlooked something here and another definition of
continuity more suitable for "hyperreals" or some computer-algebra
topology fixes this all up.
Here's a fun thing to do with our function f[x_]:=1/(1-Exp[I x]) .
try Plot[Re[f[x]],{x,0.01,20}]
On my system it look like a plot of y=0. A bug (Mathematica 7)?
And yet Re[f[0.5]] is 0.5, not zero. same for Re[f[0.01]].
Next try Plot[f[x]],{x,0.01,200}], to see some bubbles and a large oddly
positioned spike between 70 and 80.
In fact, Re[f[x]] should always be 1/2.
As for Mathematica's return of Limit[f[x],x->0] as I*Infinity, I'm
pretty sure that 1/2+I*Infinity is a better answer. I suspect it is a
mistake to "simplify" that to I*Infinity. Here's why.
Limit[f[x]-I/x,x->0] comes out 1/2.
My answer preserves the identity lim(A+B) = lim(A)+lim(B).
Mathematica's does not. [This identity holds if lim(A) and lim(B) both
exist, which apparently they do in Mathematica.)
Back to Andrzej's note.
Sin(1/z) as z->0 is not the same as Sin[z] as z->ComplexInfinity.
1/z as z->0 has a neighborhood. There is no neighborhood around
ComplexInfinity.
(I am not, incidentally, proposing that I have a complete solution to
these issues.)