Functions such as sin(x) are not considered to have limits as x approaches infinity. Sequences such as Grandi's series of 1-1+1-1+1... are not considered to have sums classically but with expanded summation definitions such as Ceasro, and Abel summation the sequence is considered to sum to 1/2. I assume that someone has come up with an equivalent expanded definition of a limit for oscillatory functions at infinity but despite searching for it I could not find any mention. I was hoping that someone could give me the key words, pointing me in the right direction to find these expanded definitions.
The reason I am interested in this is that I was considering quantum infinite potential well's with solutions of the sin(kx)² for the density function. At high energies k approaches infinity and the solution does not exist. But thinking about it more the function rapidly oscillates between 0 and 1 at this limit, for a probability density function the zero parts can be more or less ignored when they are infinitesimal in size. If I view the limit as instead going to a constant function at a non-zero value it makes physical sense as it would correspond to the classical limit of a particle moving at constant velocity in the well.
Thanks