Need the first option, whether it is correct or not.
For $\beta =1$ we have, $x_{k+1}=2^{-1+\log_2 x_k}$ .Using recurrence we have, $x_{k+1}=2^{-(k+1)}x_0$. Then $\displaystyle \lim_{k\to \infty}\left|\frac{x_{k+1}}{x_k}\right|=2^{-1}$.
So the sequence $\{x_k\}$ converges to $0$ linearly and rate of convergence is $2^{-1}$.
So the first option is incorrect. Am I correct ?