Radish pie

I recently came across a rather peculiar recipe for something called Babylonian radish pie. Intrigued, I began to follow the directions, which said I could start with any number of cups of flour.

Any number? I mean, I had to start with some flour, so zero cups wasn’t an option. But according to the recipe, any positive value was fair game. Next, I needed a second amount of flour that was 3 divided by my original number. For example, if I had started with two cups of flour, then the recipe told me I now needed 3 divided by 2, or 1.5, cups at this point.

I was then instructed to combine these amounts of flour and discard half. Apparently, this was my new starting amount of flour. I was to repeat the process, combining this amount with 3 divided by it and then discarding half. The recipe told me to keep doing this, over and over. Eventually, I’d have the proper number of cups of flour for my radish pie.

How many cups of flour does the recipe ultimately call for?

So if $x_0 \gt 0,$ with $$x_n = \frac{1}{2} \left( x_{n-1} + \frac{3}{x_{n-1}} \right),$$ for $n \geq 1.$ Let's assume that $x_n \to x^*,$ then the limit would have to satisfy $$x^* = \frac{1}{2} \left( x^* + \frac{3}{x^*} \right)$$ or equivalently $x^* = \frac{3}{x^*}$ or $x^* = \sqrt{3}.$