We have shown that the solution of the carousel-pendulum problem can be analytically found, provided one has the patience to solve a quartic equation. It is actually more conveniant to find a numerical solution, which can then be used to check the accuracy of an approximate solution. It appears that this approximate solution is valid over a wider range of ratio l∕R than initially expected. We think that this simple study should convince students that the determination of the domain of validity of an approximation is a difficult task, even if an analytical solution exists. Simply stating that a certain quantity has to be small (l∕R in our case) may be too restrictive depending on the relative error accepted on the final result. In the above problem, using l∕R = 0.7 still gives a relative error on θ smaller than 10%.