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Указав 𝑁(𝜀), вычислите пределы

The limit in question is:

\[
\lim_{n \to \infty} \frac{n^2 + 6}{n^2 - 10n + 26}
\]

### Step-by-Step Solution:

1. **Identify the highest degree term**:  
   The degree of the numerator and denominator are both 2 because \(n^2\) is the highest degree term in both expressions.

2. **Divide both numerator and denominator by \(n^2\)**:  
   To simplify the limit, divide both the numerator and denominator by \(n^2\), the highest power of \(n\).

\[
\frac{\frac{n^2 + 6}{n^2}}{\frac{n^2 - 10n + 26}{n^2}} = \frac{1 + \frac{6}{n^2}}{1 - \frac{10}{n} + \frac{26}{n^2}}
\]

3. **Take the limit as \(n \to \infty\)**:  
   As \(n\) approaches infinity, the terms \(\frac{6}{n^2}\), \(\frac{10}{n}\), and \(\frac{26}{n^2}\) approach zero:

\[
\lim_{n \to \infty} \frac{1 + \frac{6}{n^2}}{1 - \frac{10}{n} + \frac{26}{n^2}} = \frac{1 + 0}{1 + 0 + 0} = \frac{1}{1} = 1
\]

### Final Answer:
\[
\lim_{n \to \infty} \frac{n^2 + 6}{n^2 - 10n + 26} = 1
\]

Would you like further details on any step or additional explanation?

Here are some related questions you might find useful:
1. What happens when the degree of the numerator is larger than the denominator?
2. How would you approach this limit if the degrees were different?
3. Can you give an example of a limit where the result is infinity?
4. What is the significance of dividing by the highest degree term?
5. How can you apply this technique to more complex rational functions?

**Tip:** When evaluating limits at infinity, always focus on the highest degree terms in the numerator and denominator—they dominate the behavior of the function.

lim (n→∞) (n² + 6) / (n² - 10n + 26);

Learn how to evaluate the limit of the rational function (n² + 6) / (n² - 10n + 26) as n approaches infinity. This problem covers key concepts in limits, rational functions, and asymptotic behavior, suitable for advanced high school or early college-level calculus students.

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