To determine which of the given sets is infinite, let's analyze each set:

1. Set $U$:

   • $U = \{ \text{Positive real numbers less than 300} \}$.
   • This set contains all positive real numbers less than 300. However, since it is limited by the number 300, $U$ is not infinite.

2. Set $A$:

   • $A = \{ x \in \mathbb{N} \mid x \text{ is divisible by 15} \}$.
   • This is the set of natural numbers divisible by 15. Given that $U$ limits the elements to numbers less than 300, $A$ would contain numbers like 15, 30, 45, ..., 285, which is a finite sequence. Thus, $A$ is not infinite.

3. Set $B$:

   • $B = \{ 2, 4, 6, 8, \dots, 198, 200 \}$.
   • This set is clearly a finite arithmetic sequence, with a last term of 200, so $B$ is not infinite.

4. Set $C$:

   • $C =$ The set of all natural numbers divisible by 5.
   • This set includes numbers like 5, 10, 15, 20, and so on, without any upper limit. Therefore, $C$ is infinite.

5. Set $D$:

   • $D = \{ x \in \mathbb{R} \mid x < 100 \}$.
   • This set includes all real numbers less than 100. However, it does not extend to infinity because it's limited to numbers less than 100, so $D$ is not infinite.

6. Set $E$:

   • $E = \{ 3, 6, 9, 12, \dots \}$.
   • This set is the sequence of multiples of 3, and since it doesn't have an upper bound, it is infinite.

Conclusion: The infinite sets among the described ones are Set $C$ and Set $E$.

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Would you like any further details or explanations?

Here are some related questions you might find useful:

1. How do you determine if a set is infinite?
2. What is the difference between countably infinite and uncountably infinite sets?
3. Can a set of real numbers be finite?
4. How does the concept of a universal set limit the elements of a subset?
5. What are the properties of arithmetic sequences?
6. How do you find the number of elements in a finite set?
7. What are natural numbers and how do they differ from real numbers?
8. How can we describe the intersection of infinite sets?

Tip: Always check the definition and limits of a set to determine if it is finite or infinite.