The following theorem appears in secondary school pure mathematics textbook.
Let be an elementary function. If is well-defined, then
Fine. Then a student, chan, asked,
How about ? Is it an elementary function?
Urm, to me, the definition of elementary function is very vague, and I replied to the student, yes, power function is elementary, right?
Then, another question turned up. The following limit DOES NOT exist
(because the left-hand limit does not exist) it is NOT equal to !
So, what is wrong?
The way of introducing elementary functions to students, in most of the cases, is giving examples, not by definition. When students asking what is NOT an elementary function, it is easy to give, say, , , , etc. If telling students the stuff just mentioned, just like telling junior form students that is not a rational number. May be I need to study Risch algorithm.
Anyway, I believe that is elementary, so what should be refined in the so-called theorem quoted?
Instead of “ is well-defined" as mentioned in the theorem, should I refine it as “ is well-defined on the neighbourhood of “? Just a question, no conclusion yet.