Diagonalization & Matrix Powers · Medium problem

Let $A = PDP^{-1}$ with $P = \begin{pmatrix}1 & 1\\1 & 2\end{pmatrix}$ and $D = \begin{pmatrix}1 & 0\\0 & \tfrac12\end{pmatrix}$. As $n \to \infty$, $A^n$ approaches a limit matrix $L$. Find the entry in row 1, column 1 of $L$.