The image presents a solution related to the behavior of a recursive sequence, likely in the context of mathematical analysis or calculus. Here's a breakdown of what it contains:

1. Boundedness of the Sequence:

The equation for the recursive sequence is given as:

$x_{n+1} = \sqrt{2 + x_n} \leq 2$

This means that the sequence $x_n$ is bounded above by 2.

2. Limit of the Sequence:

The analysis continues by determining the limit $L$ of the sequence as $n \to \infty$. Assuming the sequence converges to a limit $L$, it satisfies the equation:

$L = \sqrt{2 + L}$

Squaring both sides leads to:

$L^2 = 2 + L$

Rearranging this:

$L^2 - L - 2 = 0$

Factoring this quadratic equation:

$(L - 2)(L + 1) = 0$

The two possible solutions are $L = 2$ and $L = -1$. Since the sequence is bounded above by 2 and presumably non-negative, the solution $L = -1$ is rejected. Therefore, the limit of the sequence is:

$L = 2$

Conclusion:

The limit of the sequence is $2$, and the sequence is bounded.

Would you like more details or explanations on any specific part? Here are 5 related questions you could explore:

1. What types of sequences have bounded limits?
2. How do you prove that a recursive sequence converges?
3. Could the sequence $x_n = \sqrt{2 + x_n}$ ever have a negative limit?
4. What are the conditions for squaring both sides of an equation in recursive sequences?
5. Can this type of problem be solved using a different method, such as fixed-point iteration?

Tip: When working with recursive sequences, always check for boundedness and monotonicity to verify convergence.