I took the liberty of circulating to our Math Dept Andrej Bauer's message which included the example
Limit[((1 + 4 x^2)^(1/4) - (1 + 5 x^2)^(1/5))/(a^(-x^2/2) - Cos[x]), x -> 0]
In retaliation my colleague Alf van der Poorten commented as below. --Ross ------------------------------------------------------------------------ - Dear Ross, I've long said that "mathematics ain't about getting it right but about not getting it wrong". Our students of course think that we teach them methods/algorithms/recipes for getting it right --- indeed, we often give them partial reward just for seeming to have started to attempt to use some reasonable way that might possibly have led them to a correct answer. In that context looking it up or having a machine look it up (or asking/copying from a wise friend) is a perfectly fine recipe for maybe getting it right. The trick is verifying that it is indeed very likely right (or, occasionally, actually \emph{is} right) and much better yet, noticing that it is wrong or that there are grounds for suspicion warranting their consulting a dinkum mathematician to get the good oil. By the way, the particular limit cited didn't tempt me towards error at all: Almost automatically I scribbled $(1+4x^2)^{1/4}=1+x^2+\ldots \,$, and so on, realising a line or so later that I had better warm up those $\ldots\,$ to $-\frac32x^4+\ldots\,$, and so on. I hadn't realised that I was checking whether some zero is isolated and am sorry to learn that I might have been. Ideally our students will learn from us to see when they've surely got it wrong and, more subtly, when they might well possibly have got it wrong ("Yup, it does indeed work for $a=1$ but, oops! Why is there an unspecified $a$ at all? I'd better phone a mathematician to ask --- hoping she does something sensible such as reminding me of Taylor expansions rather than waffling about isolated zeros."). Unfortunately, as always, distinguishing right from wrong requires quite a bit of knowledge (both of what's right and of what isn't) that students are often slow in acquiring. ------------------------ Baseball and Quantum Physics Umpire: "Some is balls and some is strikes, but until I calls 'em they ain't nothing." Centre for Number Theory Research 1 Bimbil Place, Killara NSW 2071 +61 2 9416 6026