On 01/28/2010 06:44 PM, Richard Hennessy wrote:
> I think this is a bug after reading your detailed explanation. The
> boundary of the limit f(x) as x-> a is not the same as the value f(a)
> in limits. So limit(log(x),x,0) = minf.
> $infinity means complex infinity which is not correct. The answer is
> real minf.
> Rich
>
>
> On Tue, Jan 26, 2010 at 2:44 PM, Sheldon Newhouse <sen1 at math.msu.edu
> <mailto:sen1 at math.msu.edu>> wrote:
>
> 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
>
>
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>
I did not know there was a "minf".
So, yes, assuming that "log(x)" is the log of the real positive number
x, then
limit(log(x),x,0) = minf.
The same bug propagates to some, but not all functions.
(%i1) display2d: false;
(%o1) false
(%i2) limit((1-x)*log(x),x,0);
(%o2) infinity
(%i3) limit(cos(x)*log(x),x,0);
(%o3) minf
-sen