Absolute error minimal supremum

This is definition from book:

1. Number $\displaystyle \Delta (x') = \left| {x - x'} \right|$ represents absolute error of number $\displaystyle x'$

Then it says that in real life is often impossible to find out value of absolute error $\displaystyle \Delta (x') $ but it is possible to find out smallest supremum of $\displaystyle \Delta (x') $.

So $\displaystyle \Delta (x') = \left| {x - x'} \right| \le \Delta _{x'} $

$\displaystyle \Delta _{x'} $ is smallest supremum.

Can someone explain me on some example this smallest supremum?