Chapter 11.P, Problem 4P

Let $\left\{P_n\right\}$ be a sequence of points determined as in the figure. Thus $\left|A{P}_1\right|=1$, $\left|P_n{P}_{n+1}\right|=2^{n-1}$, and angle $A{P}_n{P}_{n+1}$ is a right angle. Find ${{\displaystyle \lim }}_{n\to \infty }\angle P_{n}A{P}_{n+1}$.