Answer
$$\lim\limits_{n \to \infty}\angle P_nAP_n+1=\frac{\pi}{3}$$
Work Step by Step
We have $\lim\limits_{n \to \infty}(\frac{2^{n-1}}{\sqrt {4^n+\frac{2}{3}}})=\sqrt {\lim\limits_{n \to \infty}(\frac{3.2^{(-2)}}{1+2^{1-2n}}})=\frac{\sqrt 3}{2}$
Hence,
$$\lim\limits_{n \to \infty}\angle P_nAP_n+1=\frac{\pi}{3}$$
