The image shows a mathematical task where you are asked to select the correct statements. The task appears to involve the behavior of functions, limits, and asymptotic analysis. Here's a brief summary of the given statements:

1. The first statement seems to discuss the asymptotic behavior of functions $\alpha(x)$ and $\beta(x)$ as $x \to a$ and limits involving Landau notation $o(\alpha(x))$ at $x \to a$.

2. The second statement refers to a composite function $f(\varphi(t))$ and claims that if the function $\varphi$ has a limit at some point $t_0$, then so does the composite function.

3. The third statement discusses the existence of the limit of the function $y = f(x)$ as $x \to x_0$.

4. The fourth statement adds conditions about the limit of the composite function under the assumption that $f(\varphi(t))$ does not equal a specific point in a punctured neighborhood.

5. The final two statements discuss the asymptotic behavior of a small function $\alpha(x)$ as $x \to x_0$ or as $x \to \infty$, with one involving the sine function and the other involving a quadratic asymptotic estimate.

Would you like to go through the analysis of each of these statements in more detail, or clarify anything in particular?

Here are 5 related questions to explore further:

1. What is the definition of asymptotic behavior and how is it used in limits?
2. How does Landau notation ($o$, $O$) describe the behavior of functions at a given point?
3. How does the behavior of composite functions depend on their components' limits?
4. What are the conditions for the existence of limits in punctured neighborhoods?
5. How is the concept of an infinitesimally small function used in calculus and analysis?

Tip: When evaluating limits of composite functions, always verify if both the inner and outer functions have well-defined limits at the relevant points.