problem: What is the numerical value of \[ x = \frac{1}{2-\frac{1}{2-\frac{1}{2-\ldots}}} ?\]
solution: Noticing that  \[ \frac{1}{x} = 2 - \frac{1}{2-\frac{1}{2-\frac{1}{2-\ldots}}} = 2 - x, \] we have only to solve the quadratic equation $x^2 - 2x +1 = (x-1)^2 = 0$. Thus we see that $x = \boxed{1}$.