$a_1 =2$ and $a_{n+1}= \frac{2a_n +3}{a_n +2}$ the recursive sequence , converges?, and if yes, show it to which converges I do not know what to do, I try showing the first elements to see if they were behaving in some way, but no

Let $f(x)=(2\,x+3)/(x+2)$. There is a unique point $x>0$ such that $f(x)=x$, namely $x=\sqrt3$. Since $2>\sqrt3$, I will focus on the interval $[\sqrt3,\infty)$. On that interval $f(x)\ge\sqrt3$ and $f(x)\le x$, with equality only for $x=\sqrt3$. Use this to show that $a_n$ is decreasing and bounded below by $\sqrt3$. This implies that $a_n$ is convergent. Take limits in the equality defining $a_{n+1}$ and find the limit.
