Let $\left\{a_{n}\right\}$ be a convergent sequence defined by $a_{1}=1$, and $a_{n+1}=\frac{a_{n}+2}{2 a_{n}+5}, 0< a_{n}< 5$
Find the limit of the sequence.

Step 1 ：Let \(\left\{a_{n}\right\}\) be a convergent sequence defined by \(a_{1}=1\), and \(a_{n+1}=\frac{a_{n}+2}{2 a_{n}+5}\), where \(0

Step 2 ：Since the sequence is convergent, the limit of the sequence, say \(L\), should satisfy the equation \(L = \frac{L+2}{2L+5}\)

Step 3 ：Solving this equation gives two solutions: \(-1 + \sqrt{2}\) and \(-1 - \sqrt{2}\)

Step 4 ：However, we are given that \(0 < a_{n} < 5\), so the limit of the sequence must be positive and less than 5

Step 5 ：Therefore, the limit of the sequence is \(-1 + \sqrt{2}\)

Step 6 ：Using python code to simplify the final answer, we get \(\boxed{-1 + \sqrt{2}}\)