$\left\{\begin{array}{l} u_1=2\\ u_{n+1}=\dfrac{u_n^2+4}{4}\end{array} \right.\\ u_2=\dfrac{u_1^2+4}{4}=\dfrac{2^2+4}{4}=\dfrac{8}{4}=2\\ u_3=\dfrac{u_2^2+4}{4}=\dfrac{2^2+4}{4}=\dfrac{8}{4}=2\\ ........\\ u_{n+1}=\dfrac{u_n^2+4}{4}=\dfrac{2^2+4}{4}=\dfrac{8}{4}=2\\ \Rightarrow u_n=2 \, \forall n \in \mathbb{N^*}\\ \lim u_n=2 \,$