What is the numerical value of \[ x = \frac{1}{2-\frac{1}{2-\frac{1}{2-\ldots}}} ?\]
One way to approach this problem is to notice that the fraction has a self-similar structure: the denominator of each fraction is 2 minus another fraction of the same form. This suggests that we can write x as a function of itself: \[ x = \frac{1}{2-x} \] To solve for x, we can cross-multiply and get \[ x(2-x) = 1 \] or \[ x^2 - 2x + 1 = 0 \] This is a quadratic equation, which we can solve by factoring or using the quadratic formula. Factoring, we get \[ (x-1)(x-1) = 0 \] So x = 1 is the only solution. 