Example 10. Solve the equation
$$
\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+x}}}}=x .
$$
When solving this problem, you can guide students to use conjecture and recursive methods to explore the solution path. By observation, one can conjecture: Is $x$ $=\sqrt{3+x}$? After several iterations, we get:
$$
\begin{aligned}
x & =\sqrt{3+x}=\sqrt{3+\sqrt{3+x}} \\
& =\sqrt{3+\sqrt{3+\sqrt{3+x}}} \\
& =\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+x}}}} .
\end{aligned}
$$

Therefore, it is sufficient to solve the simple irrational equation $x=\sqrt{3+x}$.