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The image shows a series of limit problems with instructions in Russian, asking to "Select all the correct options." I will evaluate each limit step by step:

1. **\( \lim_{n \to \infty} \frac{n^2 + n + 2}{n^3} = 1 \)**  
   **Solution:**  
   The dominant term in the numerator is \( n^2 \), and in the denominator, it is \( n^3 \).  
   So, \( \frac{n^2 + n + 2}{n^3} \approx \frac{n^2}{n^3} = \frac{1}{n} \to 0 \).  
   Therefore, **this is false**.

2. **\( \lim_{n \to \infty} \left( \sqrt{n} - \sqrt{n+1} \right) = \infty \)**  
   **Solution:**  
   We can simplify this by multiplying by the conjugate:  
   \[
   \sqrt{n} - \sqrt{n+1} = \frac{(\sqrt{n} - \sqrt{n+1})(\sqrt{n} + \sqrt{n+1})}{\sqrt{n} + \sqrt{n+1}} = \frac{-1}{\sqrt{n} + \sqrt{n+1}}
   \]
   As \( n \to \infty \), this expression tends to \( 0 \).  
   So, **this is false**.

3. **\( \lim_{n \to \infty} \left( 1 + \frac{1}{n^3} \right)^{n^2} = e \)**  
   **Solution:**  
   This is a standard form of a limit resembling the expression for \( e \). The expression approaches \( e \) because for large \( n \), the term \( \frac{1}{n^3} \) tends to 0 slowly.  
   Thus, **this is true**.

4. **\( \lim_{n \to \infty} \frac{n^2 + n + 2}{n^3} = 0 \)**  
   **Solution:**  
   As discussed in the first case, this limit is indeed \( 0 \), not \( 1 \).  
   Therefore, **this is true**.

5. **\( \lim_{n \to \infty} \left( \sqrt{n} - \sqrt{n+1} \right) = 0 \)**  
   **Solution:**  
   From the second case, we determined that the expression tends to 0.  
   So, **this is true**.

6. **\( \lim_{n \to \infty} \left( 1 + \frac{1}{n^3} \right)^{n^2} = 1 \)**  
   **Solution:**  
   This does not approach 1, as the expression behaves similarly to the one for \( e \).  
   Hence, **this is false**.

### Correct answers:
- **\( \lim_{n \to \infty} \left( 1 + \frac{1}{n^3} \right)^{n^2} = e \)** (true)
- **\( \lim_{n \to \infty} \frac{n^2 + n + 2}{n^3} = 0 \)** (true)
- **\( \lim_{n \to \infty} \left( \sqrt{n} - \sqrt{n+1} \right) = 0 \)** (true)

Do you want more details or have any other questions?

Here are 5 related questions:
1. How can limits help in evaluating indeterminate forms?
2. What techniques can simplify limits involving square roots?
3. How does L'Hopital's Rule work in solving limits?
4. How is the concept of "dominant terms" useful in evaluating limits?
5. How do limits behave for exponential forms like \( e^x \)?

**Tip:** Always check for indeterminate forms like \( \frac{0}{0} \) or \( \infty - \infty \) when solving limits, as they often require special techniques like L'Hopital's Rule or simplification through algebraic manipulation.

Выберите все верные варианты среди предложенных пределов.

Learn how to solve advanced limit problems involving polynomials, square roots, and exponential growth. This problem is suited for university-level calculus students, exploring concepts like asymptotic behavior and exponential limits.

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