In our course notes, it says: "The CLT states that if random samples of size $n$ are repeatedly drawn from any population with mean $\mu$ and variance $\sigma^2$, then when $n$ is large the distribution of the sample means will be approximately normal."
I bolded "repeatedly" because that's the part I'm confused about. In the examples and the homework problems, it seems that even when we have $n$ supposedly large enough ($>30$), we're still only talking about a single sample of size $n$.
For example, one of the problems states: "Suppose a population has mean $\mu=5$ and standard deviation $\sigma = 2$, then supposed a random sample of size 38 is selected. What is the probability that the sample mean is between $4$ and $6$?"
Then we go on to use the CLT to solve the problem, supposedly because $n > 30$ allows us to use the CLT. But from that previous paragraph in italic I quoted, it seems that the CLT only works when many many samples of size n are drawn from a population, even when $n > 30$.
Here we're only randomly selecting a single sample, yet it seems to be good enough to use the CLT. Why is that?
EDIT: When I say "all CLT problems" I mean all the ones I ran into, including the famous swan problem, in which we randomly select $n>30$ swans once.