At the risk of repeating myself, I can refer you to a paper that
suggests this fits within
a framework of interval arithmetic.
e.g. interval(-inf, inf) means --- there is a real number but we
don't know anything about its value.
this is perhaps indefinite (IND)
an empty interval (we can choose one to be canonical) that has nothing
in its interior means
--- there is no number that this could be, which is perhaps undefined (UND)
http://www.cs.berkeley.edu/~fateman/papers/interval.pdf
There are what I consider misuses of intervals regarding limits that are
present in Mathematica and Maple,
eg..
limit(sin(x),x,inf) ?=? interval(-1,1).
This leads to either erroneous answers to limit problems and things
related to limits, or to a substantially
overloaded notion of "equality". Like any interval that contains 0 is
equal to 0.
Whether this can be put together in a computer algebra system to be more
consistent is possible, but
consistency in general is tough, and certainly not achieved now.
There are many examples in which, for some procedure f,
f(a) is different from subst(a,x, simplify(f(x)) )
where x is some arbitrary symbol and a is some particular value.
RJF