Limit of an iterated function system

Question

Consider the function $h(x) = 2x(1-x).$ My goal is to find $$\lim_{n \to \infty} h^n \left ( \frac{1}{4} \right ),$$ where $h^n (x_0)$ is the $n^{\text{th}}$ iterate of the function $h(x)$ at the point $x_0.$ I have that $$h^0 \left ( \frac{1}{4} \right ) = \frac{3}{8} = .375.$$ $$h^1 \left ( \frac{1}{4} \right ) = \frac{3}{8} \left ( \frac{5}{4} \right ) = .46875.$$ $$h^2 \left ( \frac{1}{4} \right ) = \frac{3}{8} \left ( \frac{85}{64} \right ) = .4980.$$ It appears that we are approaching $\frac{1}{2}$. But, how do I prove this? I do not see an obvious closed form solution for the limit.