Let $\left[ \begin{matrix} 2 & 1 \\ 1 & 0 \end{matrix} \right]^n=\left( a_{ij}(n) \right)$

If $\left( \displaystyle \lim_{n \to \infty} \dfrac{a_{12}(n)}{a_{22}(n)} \right)^2=\sqrt{A}+\sqrt{B} \quad \quad \left( A,B \in \mathbb{N}\right)$

then find the value of $A+B$.

**Notation**: $a_{ij}(n)$ denotes the element in the $i^{\text{th}}$ row and $j^{\text{th}}$ column of matrix $A$.