Ross Street wrote:
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
I don't quite understand who is retaliating for what in the posted comment. Even though this is not strictly speaking category theory, I will take the further liberty to tell you the whole story. The above limit cropped up when I told my students in "Computer Science I" to bring examples of problems they do in Analysis and Algebra. The idea was to solve them using Mathematica. Someone bought the above limit, except that instead of the parameter 'a' he had the constant 'e' (base of natural logarithm). Now, as it happens, in Mathematica this constant is written 'E' instead of 'e'. The student of course typed in 'e' which Mathematica took as a "free parameter" and gave the "generic" answer 0, while the correct answer was 6. It took a while to explain the whole mess. I suspect the only thing the students learned was that (a) the limit is stupid, (b) Mathematica is stupid and (c) the teacher is a pedantic fanatic. By the way, to see what is going on in the above limit, draw a family of functions for various values of 'a' approaching and passing through a=e. All will be clear as you see two "waves" passing through each other. I guess I am trying to point out that current Computer Alegbra Systems are very tricky to use _correctly_. In Mathematica's defense it should be said that it also found a couple of errors where the Analsyis teacher did not (one involved cancelling "a common factor of b" between b^2 and sqrt(b^2) for a real parameter b). Andrej