This is Newton iteration for $x^2-6=0$. For every positive x, AM–GM gives $x/2+3/x\geq\sqrt6$. Thus x₁ ≥ √6. Once $x_n\geq\sqrt6$, $x_{n+1}-x_n=(6-x_n^2)/(2x_n)\leq0$, so the sequence decreases and is bounded below. Its positive limit solves $L=L/2+3/L$, hence L = √6. Therefore C.
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