the probability density for a particle in a box is an oscillatory function even for very large energies. Explain how the classical limit of a constant probability density that is independent of position is achieved

This is achieved for the specific case when high quantum number with low resolution is present.

Step-by-step explanation:

In Quantum Mechanics, the probability density defines the region in which the likelihood of finding the particle is most.

Now for the particle in the box, the probability density is also dependent on resolution as well so for large quantum number with small resolution, the oscillations will be densely packed and thus indicating in the formation of a constant probability density throughout similar to that of classical approach.