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## 'What Do Functions Do for Tiny X?' printed from http://nrich.maths.org/

Archimedes, Bernoulli, Copernicus and de Moivre each try to think
of a function which will be small near to the origin. Their choices
are:

$$ A(x) = \sin(x) \quad B(x) = 1 - \cos(x) \quad C(x) = \log(1+x)
\quad D(x) = 1 - \frac{1}{1-x} $$

All of these functions equal zero when x is exactly zero, but the
friends want to investigate how small their functions are when x is
small but not exactly zero.

Use a spreadsheet to investigate these curves graphically for
smaller and smaller values of x (you may like to try graphs with x
ranging from -1 to 1, -0.1 to 0.1, -0.01 to 0.01 to begin with).
Before you start, you might like to try to guess what will happen
close to the origin in each case.

As you zoom in the scale what do you notice happening to the graphs
in each case? What similarities do the functions have as we zoom
in? What differences do they exhibit?

Can you predict the shapes of the graphs for x between -0.000001
and 0.000001? Test your predictions.

For each function can you write a simple polynomial approximation
with 1 or 2 terms for the shape of each curve for very small values
of x?