Remember that the problem was to find the equivalent resistance of this thing with $R_1 = R_2 = 1$:

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We know the equivalent resistance must exist, because it can’t be less than zero, and every time we add more resistors in series the resistance decreases, so we have a bounded monotonic series.

Replacing the back part of the circuit with its equivalent resistance, we get a diagram like this:

Now find the equivalent resistance to $x$.

$x = 1 + \left(\frac{1* x}{1 + x}\right)$

which simplifies to

$x = \frac{1 + \sqrt{5}}{2}$,

which is the golden ratio.