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## 'How Does Your Function Grow?' printed from http://nrich.maths.org/

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 $$B(n) = n^{100}$$
- Copernicus takes powers of 100 $$C(n) = 100^n$$
- 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 $$D(n) = n\times (n-1)\times
(n-2)\times \dots \times 2\times 1$$

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.