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To solve the limit problem presented in the image, we need to apply the algebra of limits and evaluate the given expression step by step. The problem is asking us to prove that:

\[
\lim_{x \to a} \frac{x^2 + x + 1}{x^4 + 1} = \frac{a^2 + a + 1}{a^4 + 1}
\]

### Step-by-Step Solution:

1. **Substitute the Limit Expression:**

   The limit of a rational function can be found by directly substituting \(x = a\) into the expression, provided that the denominator does not become zero. The given limit is:

   \[
   \lim_{x \to a} \frac{x^2 + x + 1}{x^4 + 1}
   \]

   Since \( \lim_{x \to a} x = a \), we can substitute \( x = a \) directly into both the numerator and denominator:

2. **Numerator Substitution:**

   The numerator is \( x^2 + x + 1 \). Substituting \( x = a \), we get:

   \[
   a^2 + a + 1
   \]

3. **Denominator Substitution:**

   The denominator is \( x^4 + 1 \). Substituting \( x = a \), we get:

   \[
   a^4 + 1
   \]

4. **Conclusion:**

   Thus, after substitution, the limit becomes:

   \[
   \lim_{x \to a} \frac{x^2 + x + 1}{x^4 + 1} = \frac{a^2 + a + 1}{a^4 + 1}
   \]

This proves that the limit holds as given in the problem.

### Key Insights:

- This is a direct application of the **limit substitution property**, where \( \lim_{x \to a} f(x) = f(a) \), provided that both the numerator and denominator are continuous at \(x = a\) and the denominator does not become zero.

Would you like more details or further clarifications?

### 5 Related Questions:
1. What happens if the denominator becomes zero in a limit problem?
2. How does continuity play a role in evaluating limits directly by substitution?
3. Can limits always be evaluated by direct substitution for all types of functions?
4. What are some common techniques used to evaluate limits when direct substitution doesn't work?
5. How do we apply L'Hôpital's Rule to limits that result in indeterminate forms?

### Tip:
When directly substituting values into a limit, always ensure that the function is continuous at that point, and check if the denominator is non-zero to avoid undefined expressions.

Usando álgebra de límites y que \(\lim_{x \to a} x = a\), probar que \( \lim_{x \to a} \frac{x^2 + x + 1}{x^4 + 1} = \frac{a^2 + a + 1}{a^4 + 1} \).

Learn how to prove that \( \lim_{x \to a} \frac{x^2 + x + 1}{x^4 + 1} = \frac{a^2 + a + 1}{a^4 + 1} \) using the algebra of limits and the concept of direct substitution. This problem involves key mathematical concepts such as the algebra of limits and continuity.

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