Sequence converging to multiple solutions of an equation

Question

I'm trying to understand the question of type: Which of the reccurent sequences below converges to any solution of the equation $x\cdot{\sin(x)=1}$:

$x_{n+1} = \arcsin(1/x_{n}), x_0 = 0$

$x_{n+1} = 1/\sin(x_n), x_0 = 1$

$x_{n+1} = 1/\sin(x_n), x_0 = 0$

$x_{n+1} = \cos(x_n), x_0 = 1$

So what is exactly asked here? The equation has 4 solutions, as far as I know if a sequence converges it can only converge to one limit (subsequence can converge to multiple limits but that's not the case).

I'm not asking for the answer to the question, I'm trying to understand what is exactly asked here, basically.

Answer 1

I believe the question means to ask the following:

1. Let $x_n$ be a sequence defined by $x_0=0$ and $x_{n+1}=\arcsin(x_n)$. Does a limit $\overline{x}=\lim_{n\to\infty}x_n$ exist, and if so, do we have $\overline{x}\sin(\overline{x})=1$?
2. Let $x_n$ be a sequence defined by $x_0=1$ and $x_{n+1}=1/\sin(x_n)$. Does ...
3. etc.