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Geometry Level 4

A circle \(C_0\) with radius 1 touches both the axes and line \(L_1\) that passes through point \(P(0,4)\) and cuts the \(x\)-axis at \((x_1, 0)\).

Another circle \(C_1\) is drawn touching the \(x\)-axis, line \(L_1\) and another line \(L_2\) that passes through point \(P\) and cuts the \(x\)-axis at \((x_2, 0)\) and this process is repeated \(n\) times.

Find \(\displaystyle \lim_{n \to \infty} \frac {x_n}{2^n} \).

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