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The given problem asks to find the expressions that oscillate or alternate between different values as \( n \) increases without bound.

Let's analyze each expression one by one:

1. **\( ㉠: \{ 3n + 1 \} \)**  
   This represents the fractional part of \( 3n + 1 \), which keeps repeating for different \( n \). The fractional part cycles depending on \( n \), but it doesn't "alternate" between values like \( (-1)^n \). Instead, it will continue repeating within a certain range but not strictly oscillating in the way the problem suggests.

2. **\( ㉡: \left\lfloor \frac{2}{\sqrt{n}} \right\rfloor \)**  
   This is the floor function applied to \( \frac{2}{\sqrt{n}} \). As \( n \to \infty \), this expression tends to zero, and since the floor function results in integer values, it doesn't oscillate. Instead, it gradually decreases and eventually stays at 0 for sufficiently large \( n \).

3. **\( ㉢: (-1)^n \)**  
   This expression alternates between \( 1 \) and \( -1 \) as \( n \) increases. So this expression clearly oscillates.

4. **\( ㉣: (-3)^n \)**  
   As \( n \) increases, this expression also alternates but with increasing magnitude. For even \( n \), the result is positive, and for odd \( n \), it is negative. Hence, it oscillates as well, but with increasing absolute value.

### Conclusion:
- \( ㉠ \) does not exhibit clear oscillation.
- \( ㉡ \) gradually approaches 0.
- \( ㉢ \) oscillates between 1 and -1.
- \( ㉣ \) oscillates but increases in magnitude.

The correct answer is \( ㉢ \) and \( ㉣ \), which corresponds to option **(3)**.

Do you want further details or have any other questions?

### Expanding questions:
1. How does the floor function behave as \( n \to \infty \)?
2. What is the fractional part function and how does it cycle for large \( n \)?
3. Can we analyze the behavior of powers of negative integers for large \( n \)?
4. How does the expression \( (-1)^n \) behave in general sequences?
5. Why doesn't the expression \( \{3n+1\} \) truly oscillate?

### Tip:
To understand oscillating sequences, always check for alternating signs or periodic behavior.

다음 보기의 수열 중에서 n이 한없이 커질 때, 발생하는 것을 모두 고른 것은?

This problem focuses on analyzing different sequences as n approaches infinity, determining which sequences oscillate and which approach fixed values. Concepts include oscillating sequences like (-1)^n, fractional parts, and floor functions. Suitable for high school students.

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