Quod Erat Demonstrandum


Just a question about limit of elementary function

Filed under: HKALE,Pure Mathematics — johnmayhk @ 10:31 上午

The following theorem appears in secondary school pure mathematics textbook.

Let F(x) be an elementary function. If F(a) is well-defined, then

\lim_{x \rightarrow a}F(x) = F(\lim_{x \rightarrow a}x) = F(a)

Fine. Then a student, chan, asked,

How about F(x) = \sqrt{x}? 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

\lim_{x \rightarrow 0}\sqrt{x},

(because the left-hand limit does not exist) it is NOT equal to \sqrt{0}!

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, erf(x), \Gamma(x), Si(x), etc. If telling students the stuff just mentioned, just like telling junior form students that \pi is not a rational number. May be I need to study Risch algorithm.

Anyway, I believe that \sqrt{x} is elementary, so what should be refined in the so-called theorem quoted?

Instead of “F(a) is well-defined" as mentioned in the theorem, should I refine it as “F(x) is well-defined on the neighbourhood of a“? Just a question, no conclusion yet.


1 則迴響 »

  1. Hello!
    Very Interesting post! Thank you for such interesting resource!
    PS: Sorry for my bad english, I’v just started to learn this language ;)
    See you!
    Your, Raiul Baztepo

    迴響 由 RaiulBaztepo — 2009/03/31 @ 2:40 上午 | 回應

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