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The expression you provided is a continued fraction, which can be written as:

\[
x = 3 - \cfrac{13}{6 + \cfrac{4}{3 - \cfrac{13}{6 + \cdots}}}
\]

This is a periodic continued fraction, and typically, such expressions converge to a particular value. In this case, we aim to find the maximum value of \( x \), which often corresponds to the value that the continued fraction converges to.

### Steps to approach:
1. Since the fraction is periodic, we can assume that as the continued fraction repeats, the value stabilizes. We can denote this repeating value as \( y \), so that:

\[
y = 3 - \cfrac{13}{6 + \cfrac{4}{y}}
\]

2. Now solve this equation for \( y \), which will give the value of \( x \).

Let me solve this equation to find the value of \( y \).The solutions to the equation are \( y = -\frac{4}{3} \) and \( y = \frac{3}{2} \).

Since we are asked to find the maximum value of \( x \), we take the larger solution:

\[
x = \frac{3}{2}
\]

Thus, the maximum value of \( x \) is \( \frac{3}{2} \).

Would you like further details on how this was solved, or do you have any other questions?

Here are some related questions to explore:
1. What is the general form of a continued fraction?
2. How do periodic continued fractions typically converge?
3. Can continued fractions represent irrational numbers?
4. How would the equation change if a different pattern appeared in the continued fraction?
5. What are some applications of continued fractions in real-world problems?

**Tip**: Periodic continued fractions often arise when solving quadratic equations, and they can represent solutions like square roots of non-perfect squares.

If x = 3 - 13 / (6 + 4 / (3 - 13 / (6 + ...))), find the maximum value of x.

This problem explores the periodic continued fraction x = 3 - 13 / (6 + 4 / (3 - 13 / (6 + ...))). Learn how to find the maximum value of x using algebraic techniques. Ideal for students in Grades 11-12, this solution covers concepts such as continued fractions, periodic sequences, and quadratic equations.

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