On 01/26/2010 01:49 PM, Raymond Toy wrote:
> What is limit(log(x),x,0)? Maxima currently returns infinity. I
> suppose that's correct, but with a default domain of real, shouldn't
> this be undefined?
>
> Ray
>
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>
Read your mail more closely this time, so here is a more detailed
discussion.
The limit of a function defined on a set A could be defined for points x
in A or also for points x in the boundary of A.
Suppose f is defined from a set A to a set B. One needs some notion
of closeness of points in A and B. This is usually defined by means of
a metric (or distance function) or a topology.
To stay with the log function, let us assume that we are dealing with
A, B subsets of the real numbers.
Intuitively, the concept
lim(f(x), x, a) = b should mean that as
x approaches a and x is in A, then f(x) approaches b.
The object b need not be in B, it could be in the boundary of B if B
is in some larger set for which the boundary is defined.
In the case of plus or minus infinity, one defines the set of extended
real numbers bar{R} which is R union {-infinity, +infinity}.
The precise definition of
limit(f(x),x->a) = -infinity is that
for a point a in the boundary of A (the domain of f) and
for any real number K there is a neighborhood U of a such that if x
is in U intesect A, then f(x) < K.
So, you see that with this definition, limit(log(x),x->0) = -infinity
Similarly, one can define limit(f(x), x->a) = infinity by repacing
"f(x) < K" with "f(x) > K".
HTH,
-sen