This image appears to present a list of functions and asks to identify which ones are "infinitely small" as $x \to x_0$, with several options provided.

Let me evaluate each function:

1. $\alpha(x) = \frac{1}{x}$ when $x_0 = 0$: As $x$ approaches 0, this function diverges, going to infinity. Hence, this is not infinitely small.

2. $\beta(x) = \frac{3}{x^2}$ when $x_0 = \infty$: As $x$ approaches infinity, this function tends to 0. This qualifies as an infinitely small function at $x_0 = \infty$.

3. $\gamma(x) = \frac{\sin x}{x}$ when $x_0 = \infty$: As $x$ approaches infinity, this function tends to 0 as well, so it is infinitely small.

4. $\delta(x) = 3x$ when $x_0 = 0$: This function tends to 0 as $x \to 0$, but not infinitely small because it does not approach 0 rapidly enough (it’s a linear function).

Thus, functions $\beta(x)$ and $\gamma(x)$ are infinitely small, corresponding to the options 2 and 3.

The correct answer would be 234.

Would you like more details on any of these calculations? Here's a set of related questions to further explore:

1. What are the criteria for a function to be considered "infinitely small" near a point?
2. How do different powers of $x$ affect whether a function is infinitely small as $x \to \infty$?
3. Can a function like $\frac{1}{x}$ ever be considered small as $x \to 0$?
4. How do trigonometric functions behave in limits like $\frac{\sin x}{x}$ as $x \to \infty$?
5. What are the general strategies for determining the limit of a function as $x$ approaches infinity?

Tip: When examining functions for limits, always check whether the function approaches a constant value, grows without bound, or shrinks towards zero.