Connection between Quantum and Classical worlds

According to the Correspondence Principle Discrete Quatum levels become Continous Classical levels when n goes to infinity, meaning that the difference between two adjacent level goes to zero.

Questions;

1)
What does infinity mean here? How for do we to go from the center of the atom that infinity is realized and the Quantum World becomes Classsical?


2)
Or in case of a quantum Hormonic Oscillator when does n go to infinity and discrete energy levels disappear?



In Hydrogen Atom

E(n) ~ 1/n(squared)
E(n-1) ~ 1/(n-1)(squared)

At what point in space the classical world starts and quatum world is finished.

Or we do not know how the two world are connected or there is no quantum world because we cannot see it. All of our experiments are purely classical and never quantum so where is the quantum world?
 
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.....At what point in space the classical world starts and quatum world is finished. ....

.... All of our experiments are purely classical and never quantum so where is the quantum world?
You pretty much have it. As you said:
"we do not know how the two world are connected"
Sure you can say;
"there is no quantum world because we cannot see it."
Just recognize that as a philosophical statement not a scientific one.
Science hopes that we should be able to understand (therefore “see”) how our world works through careful scientific observation.
We just don’t know enough currently to know how to “look” at the world we are in, so for now we find it easier to think of it as "two worlds".
 

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