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Re: piecewise vs which

Andrzej Kozlowski wrote:
> On 3 Sep 2005, at 15:06, Helen Read wrote:
>> For purposes of teaching calculus students, where we are only
>> concerned with real numbers and are not taking limits in the
>> complex plane, I would like Limit to check from both directions.
> There are two basic reasons why Limit does not do that. First,  
> variables in Mathematica are by default complex so a limit has not
> just two possible direction but infinitely many. 

I understand that. As I said, however, when teaching calculus, we are 
only concerned with real numbers. I would like something like

Assuming[x ? Reals, Limit[Abs[x]/x], x -> 0]

to take the limit from Direction->-1 and Direction->1 in the real line, 
and give an answer of Indeterminate if they are not equal.

> Finally, students learning calculus should first learn to compute  
> limits by hand. By the time they can do it well enough so that it is  
> safe to let them use Mathematica for this purpose they should also  
> know why it is better to use Series.

Of course the students should learn to compute limits by hand, but as 
soon as we begin limits they ask how to do it in Mathematica so that 
they can check their answers. I'm not going to forbid this; in fact, I 
encourage students to check their work in Mathematica. When I show them 
how to use Limit -- which comes very early in Calculus I, at a time when 
the students are not yet completely comfortable with Mathematica -- I 
have to show them examples where Limit does not give the answer they 
expect, warn them to take limits from both directions, etc. The students 
and I would be much happier if Mathematica had a built-in way to do a 
two-sided limit in the real line.


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