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.