Question

Cat este $\lim_{n \to \infty} x*a^{x+1}+1$ cand a∈(0;1)?

Answer 1

$a\in(0,1)\\\\l =\lim\limits_{x\to \infty}(x\cdot a^{x+1}+1) = \lim\limits_{x\to \infty}(a\cdot a^{x}\cdot x+1)\\ \\ a^x = t \Rightarrow x = \log_ a t \\ x \to \infty \Rightarrow t \searrow 0 \\ \\ l =\lim\limits_{t \searrow 0}}(at\log_a t+1) = \lim\limits_{t\searrow 0}}(a\log_a t^t+1) = \\ \\ = a\log_a \Big(\lim\limits_{t\searrow 0} t^t\Big)+1 = a\log_a 1+1 = 0+1 = \boxed{1}$