Why should the angular displacement of a simple pendulum not exceed 4 radian

The reason the angle should be small (4° being a reasonable though arbitrariy limit) is that you need a small angle to be able to approximate the force with an elastic one (i.e. proportional to the displacement) and end up with an harmonic oscillations. If the displacement gets too big, the harmonic approximation no longer holds and you get a complex unharmonic system that is hard to describe and does weird things. For instance, the oscillation period is no longer independent on the amplitude. Hope this helps.