Pendulum's motion is simple harmonic motion For a pendulum's motion to be simple harmonic motion (S.H.M.) is it necessary for a pendulum to have small amplitude or S.H.M. can be produced at large amplitudes as well? 
If it is really necessary for an S.H.M. to have small amplitudes then why is it? because even at large amplitudes there is restoring force pulling the pendulum toward mean position and its acceleration is directly proportional to the displacement. 
 A: As you stated, in order to have simple harmonic motion, you need to have an acceleration that is proportional to the displacement.  For a pendulum, if you work out the details, you will find that 
$\frac{d^2\theta}{dt^2} \propto -\sin(\theta)$
where $\theta$ is the angle the pendulum makes with the vertical.  For small angles, $\sin(\theta)\sim\theta$, which would then lead to simple harmonic motion.  For large angles, this approximation no longer holds, and the motion is not considered to be simple harmonic motion.
A: 
In case of pendulum motion, when the angle of displacement is large(as shown in fig.), the direction of restoring force$(mg. sin \theta)$ is not exactly in the direction of equilibrium position. But the condition of S.H.M. is the restoring force must directed to the equilibrium position in all instant. So in case of large angular displacement, this condition is violated. Hence the motion of the pendulum no more remains simple harmonic in that case.
For this cause the angular displacement of S.H.M. must be kept smaller than $4$ degree. 
A: It's just because at large angular displacements, it does not approximate the SHM of, say, a block on a spring with no friction. The restoring force is not in the direction of the displacement; therefore it does not act like SHM.
