Re: decoding inbuilt function

• To: mathgroup at smc.vnet.net
• Subject: [mg121053] Re: decoding inbuilt function
• From: Murray Eisenberg <murray at math.umass.edu>
• Date: Thu, 25 Aug 2011 07:05:30 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <201108221003.GAA22469@smc.vnet.net> <201108230948.FAA02694@smc.vnet.net> <CAO-JnY3cGaQi8qZ=8hb9sKkjb51HrbFPx+KjchnnWiQXj5iOfA@mail.gmail.com>
• Reply-to: murray at math.umass.edu

```Nonsense!

The student and I may both understand the definition of limit (which
doesn't usually help to find the limit in the first place) and even know
lots of theorems that do allow one to find a limit in various situations.

But what a proprietary system does to implement finding limits is
another matter entirely.

For example, the algorithm used may be able to discern whether a
function is a polynomial, or a composition of a list of basic built-in
continuous functions, and then just use "direct substitution" -- that
is, using that Limit[f[x],x->a] has value f[a].

You might wish to consult:

http://reference.wolfram.com/mathematica/tutorial/TheAlgorithmsOfMathematica.html

Taking derivatives is an entirely different matter! As every calculus
instructor and every calculus student realizes, you don't have to know
the definition of derivative as a limit of difference quotients in order
to calculate derivatives. You just apply in some appropriate order a
whole bunch of general rules -- addition formula, chain rule, etc. --
along with a known list of derivatives for basic elementary functions.

Of course knowing which rules to apply and in which order is the trick.
We humans have heuristics for doing that. But the way it's done by a
"computer algebra system" may differ considerably from the way we do it
with paper and pencil.

Similarly, students learn how to do indefinite integration, with lots of
heuristics involved as to what standard method to try. But a CAS may do
things in a quite different way. In fact, I believe there's a
considerable literature about that.

Now if you're simply trying to rail against the very notion of a
proprietary system, then I suggest you never use an operating system
(other than Linux, etc.), never drive a modern automobile, never use a
phone, ....

On 8/23/11 8:43 PM, Ralph Dratman wrote:
> If a student comes to your office saying she cannot seem to understand this
> "limit" thingy, I wonder if you would wish to say, "Don't feel too bad, Ms
> Liddell. No one in our department understands limits much better than you
> do! Anyway, not enough to program a computer to do them. Only Wolfram has
> the algorithm for that."
>
> This seems unconscionable to me. But then, I didn't realize taking limits
> was so difficult. In order to differentiate something, obviously you have to
> take a limit. Yet I never heard of anything one couldn't manage to
> differentiate. And then having the derivative of a function would surely
> help in taking other limits involving that function. What's the big
> difficulty?
>
> Ralph
>
>
> On Tuesday, August 23, 2011, Murray Eisenberg<murray at math.umass.edu>  wrote:
>> While you may get some hints about this, even from Wolfram Research
>> employees, I suspect that there's proprietary information involved!
>>
>> On 8/22/11 6:03 AM, student wrote:
>>> hi,
>>> can any body please help me to decode the inbuilt function LIMIT in
>>> mathematica  or can any body please tell me how the inbuilt function
>>> limit works and the logic behind it
>>> please reply me soon as i really need that
>>>
>>>
>>>
>>
>> --
>> Murray Eisenberg                     murray at math.umass.edu
>> Mathematics&  Statistics Dept.
>> Lederle Graduate Research Tower      phone 413 549-1020 (H)
>> University of Massachusetts                413 545-2859 (W)
>> 710 North Pleasant Street            fax   413 545-1801
>> Amherst, MA 01003-9305
>>
>>
>

--
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305

```

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