Quantum questions

It's hard to see exactly what you are asking. The fact that you titled this "quantum questions" when there is nothing about quantum physics in it!

"How can arbitrarily large primes be applied to limit"
I have no idea what you mean by applying anything "to limit". If you mean "finding the limit of a sequence", restricting yourself to a subset of the natural numbers won't help. What happens on the composite numbers may be key to the problem.

"Is there any connection between an infinite set limit like prime numbers and differentiation and differentiability? "
Since the former is discrete and the latter continuous, none at all.

"x(infinite set of possible variables) would change with respect to y(infinite set of possible variables) on which it has a functional relationship. "
The fact that the domain is infinite is not relevant- it has to be continuous.

"Also where does probability play specifically in mathematical analysis? "
None whatsoever. Mathematical analysis (in particular measure theory plays a major role in probability.

"probability theory what happens when the sample space is infinite ?"
Not a problem. You need to define a "probabilty density" and integrate over the relevant sets.

" I have been trying to make sense of this with fourier transform and measure theory of infinite intersections/ unions."
Your first few questions don't seem to indicate that you know even basic calculus. Learn that first!

(Hmm- discrete versus continuous- could this be--- )

 

On the topic of prime numbers, I think have similarly been thinking about how they seem to mimic eigenvalues or eigen functions or a statespace. Eigen functions are those functions that undergo no transformation when an operator is applied to them (whether this is a Hamiltonian or a momemtum operator) is largely irrelevant. Prime numbers seem to be special from this perspective. I also came across a paper that proved that you can scale Energy levels in a harmonic oscillator to fit into the distribution of prime numbers. This hints at a deeper connection between numbers in general as representing a set of discrete eigen values.