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beschreibe ausführlich den angeführten rechengang und ergänze die beiden fehlenden zeilen.

To describe the given process thoroughly and complete the two missing lines, let's walk through the steps involved in the provided solution and break them down.

### Step-by-Step Process:

#### 1. **Limit Calculation**:
   The given limit is:

   \[
   \lim_{x \to \infty} \frac{x + 5}{3x - 3} = \frac{1}{3}
   \]

   The limit is evaluated as \( x \to \infty \), so we focus on the highest powers of \( x \) in the numerator and denominator. Simplifying the expression:

   \[
   \frac{x + 5}{3x - 3} = \frac{x(1 + \frac{5}{x})}{x(3 - \frac{3}{x})} = \frac{1 + \frac{5}{x}}{3 - \frac{3}{x}} \to \frac{1}{3} \text{ as } x \to \infty
   \]

   The limit calculation shows that as \( x \to \infty \), the expression approaches \( \frac{1}{3} \).

#### 2. **Epsilon (ε) Definition and Analysis**:
   The next step involves expressing the error term \( \varepsilon \). The expression becomes:

   \[
   \frac{x + 5}{3x - 3} = \frac{6}{3x - 3} = \frac{2}{\bar{x} - 1} = \varepsilon
   \]

   Here, \( \bar{x} \) is introduced as a simplification of the expression in terms of a normalized variable.

#### 3. **Case Analysis for \( x < 1 \) and \( x > 1 \)**:
   Now, the equation is split into two cases depending on whether \( x < 1 \) or \( x > 1 \).

   **For \( x < 1 \)**:

   \[
   \frac{x - z}{|\bar{x} - 1|} = \frac{2}{\varepsilon}
   \]

   **For \( x > 1 \)**:

   \[
   \frac{x - z}{|\bar{x} - 1|} = \frac{2}{\varepsilon + 1}
   \]

#### 4. **Application of \( f(x) - a \)**:
   The function \( f(x) - a \) for the cases \( x < 1 \) and \( x > 1 \) is represented as the absolute value:

   \[
   |f(x) - a| < \varepsilon \text{ for both cases.}
   \]

#### Completing the Missing Lines:

Looking at the box where the two cases are written, the missing lines are most likely intended to complete the proof of the final result. Let's analyze and write the most likely completion:

### Missing Lines Completion:

For \( x > 1 \), the second missing line should complete the expression using the information provided:

\[
\frac{x + 5 - (x - 1)}{3x - 3} = \frac{2}{\varepsilon} = \varepsilon
\]

This represents the concluding part of the \( f(x) - a \) inequality applied for \( x > 1 \).

Similarly, for \( x < 1 \), the first missing line should show:

\[
\frac{x + 5 - (x - 1)}{3x - 3} = \frac{2}{\varepsilon + 1} = \varepsilon
\]

Thus, we now have the complete process.

### Summary:

- The limit is calculated as \( \frac{1}{3} \).
- \( \varepsilon \) is introduced to express the error terms in terms of \( \bar{x} - 1 \).
- Two cases are considered: \( x < 1 \) and \( x > 1 \), each giving a relation involving \( \varepsilon \).
- The two missing lines are completed as shown to finalize the inequality for both cases.

Would you like further details on any part of the explanation, or clarification on the steps? Here are some related questions for practice:

1. How do you simplify expressions involving limits to infinity?
2. What is the significance of epsilon (\( \varepsilon \)) in limit problems?
3. How can absolute value inequalities be used in limit proofs?
4. What are common steps for handling cases when analyzing limits?
5. How can you apply this type of analysis to a different function?

**Tip**: When simplifying limits at infinity, focus on the dominant term, usually the highest power of \( x \). This helps reduce the complexity of the expression.

Beschreiben Sie den angeführten Rechengang und ergänzen Sie die fehlenden beiden Zeilen.

This problem explores the limit of a function as x approaches infinity and involves epsilon analysis with case-by-case breakdown for x < 1 and x > 1. Includes steps to complete the missing lines for inequality proofs.

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