Question

Din variante e bac 2009 mate M1


Ex. 1 - b

Answer 1

$x_{n+1} = \dfrac{2x_n+1}{x_n+2}\\ \\ L = \lim\limits_{x\to \infty }x_n = \lim\limits_{x\to \infty }x_{n+1} = \lim\limits_{x \to \infty}} \dfrac{2x_n+1}{x_n+2} = \dfrac{2\lim\limits_{x \to \infty}x_n+1}{\lim\limits_{x \to \infty} x_n + 2}}\\ \\ \Rightarrow L = \dfrac{2L+1}{L+2} \Rightarrow L^2+2L = 2L+1 \Rightarrow L^2 = 1,\quad L > 0\\ \\ \Rightarrow \boxed{L = 1}$