Four enthusiastic mathematicians are asked to think of a function involving the number 100. The challenge is to think of the function which is biggest for big values of n

Archimedes chooses a logarithm function

A(n) = log(100n)
Bernoulli decides to take 100th powers

B(n) = n¹⁰⁰
Copernicus takes powers of 100

C(n) = 100ⁿ
and, finally, de Moivre, who likes to be different, chooses the factorial function which he claims will be quite big enough without any reference to 100 at all

Which function is biggest for large values of n? Can you determine a value beyond which you know this function will be biggest?

[Extension: To find the exact switch-over value will be difficult and will require the clever use of a spreadsheet or computer.]

What could you say if the 100s were replaced by a million? billions? Create a convincing argument to prove your results to the mathematicians.