This image contains a series of mathematical statements in Russian about limits, functions, and infinitesimal analysis. Here's a breakdown of the key parts:

1. Statement 1:

   • Functions $\alpha(x)$ and $\beta(x)$ are infinitesimally small as $x \to a$, and $\lim_{x \to a} \frac{\alpha(x)}{\beta(x)} = 0$. The statement claims: $\alpha(x) = o(\beta(x)) \text{ as } x \to a.$ This is consistent with the definition of little-o notation, where $\alpha(x)$ is asymptotically smaller than $\beta(x)$ as $x \to a$.

2. Statement 2:

   • This discusses the limit of a composite function $f(\varphi(t))$. It assumes that $\varphi(t)$ has a limit $x_0$ as $t \to t_0$, and the function $f(x)$ has a limit $b$ as $x \to x_0$. Therefore, the limit of the composite function is also $b$, that is: $\lim_{t \to t_0} f(\varphi(t)) = b.$ This is valid under the assumption of continuity and proper limits.

3. Statement 3:

   • It adds the condition that in some punctured neighborhood of $t_0$, the condition $\varphi(t) \neq x_0$ holds for $t \in \dot{U}(t_0)$. This leads to the conclusion that the limit of the composite function $f(\varphi(t))$ still equals $b$, which is correct based on limit continuity.

4. Statement 4:

   • The function $\alpha(x)$ is infinitesimally small as $x \to \infty$, and: $\lim_{x \to \infty} \frac{\sin(\alpha(x))}{\alpha(x)} = 1.$ This is correct, as the limit is a standard result for small-angle approximations where $\sin(x) \approx x$ as $x \to 0$.

5. Statement 5:

   • The function $\alpha(x)$ is infinitesimally small as $x \to x_0$, and it claims that: $\lim_{x \to x_0} \frac{\alpha^2(x)}{\alpha(x)} = 0.$ This also seems correct based on the given condition that $\alpha(x)$ tends to 0.

Would you like further details on any specific statement or have any questions?

Here are 5 questions that could extend this discussion:

1. What is the meaning of little-o notation in asymptotic analysis?
2. How does the composition of limits work in more general cases?
3. Can you provide a proof for the small-angle approximation $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$?
4. What is the difference between punctured and regular neighborhoods in limit analysis?
5. How do infinitesimals behave differently as $x \to 0$ and $x \to \infty$?

Tip: Always verify the conditions of continuity and differentiability before applying limit theorems to composite functions.