# [Maxima] limit(log(x),x,0)?

Richard Hennessy richhen2008 at gmail.com
Thu Jan 28 17:44:55 CST 2010

```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> 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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