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Re: how to explain this weird effect? Integrate

Andrzej Kozlowski <akoz at> wrote in message news:<c0kr0g$fo8$1 at>...
> On 14 Feb 2004, at 03:56, steve_H wrote:
> >
> > mathematically speaking, 1/a when a=0, is the same as Limit[1/a , a->0]
> > So, the final answer should not be different.
> >
> >

> Really? If so then for over 20 years I have teaching it all wrong, and 
> so have all other professional mathematicians for much longer. If the 
> limit of a function at a point is just the value of that function then 
> why do we need to concept of limit at all?


No need to take things personally. I am sure you are a good teacher.

But we study limits to know how a function *behaves* as it approaches 
the limit. 

We would like to know for example if the function have the same values 
as it approaches the limit from above or below. We also study limits 
to find things like if a series converges or not. These are the main
reasons for needing the limit function.

I should have been more clear. I was thinking of a well behaved smooth
continouse function f(x) at a point a, the lim f(x) as x approaches a 
from either side exist and will be f(a). What else could it be? 

No matter how you look at it, the limit of f(x)=x as x approaches 5 will be 5.

Do you know of a function that is well behaved and smooth at a point 'a', 
where  the limit of f(x) as x approaches 'a' will not be f(a) ? if so, 
please tell us about this function.

> Quite apart from that, the order of evaluation is a key issue in 
> functional programming languages and there are lots of cases where 
> Mathematica will produce different answers if you change the order of 
> evaluation even if "mathematically" you are computing the same thing.

Ok, that is my main point then. All what I was saying is that we are looking
at results that are artifacts of artificial programming side-effects, and
are not results due to mathematics proper. 

It seems one will need to learn to work around these limitations in 
computer algebra systems.


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