What is the numerical value of x: $x = \frac{1}{3-\frac{1}{3-\frac{1}{3-\ldots}}}$
+26388 What is the numerical value of \(x\):
\(x = \dfrac{1}{3-\dfrac{1}{3-\dfrac{1}{3-\ldots}}}\)
\(\begin{array}{|rcll|} \hline x &=& \dfrac{1}{3-\dfrac{1}{3-\dfrac{1}{3-\ldots}}} \\\\ x &=& \dfrac{1}{3-x} \\\\ x(3-x) &=& 1 \\\\ 3x-x^2 &=& 1 \\\\ \mathbf{x^2-3x+1} &=& \mathbf{0} \\\\ x &=& \dfrac{3\pm \sqrt{9-4*1} }{2} \\\\ x &=& \dfrac{3\pm \sqrt{5} }{2} \\\\ \mathbf{x} &=& \mathbf{\dfrac{3-\sqrt{5} } {2} } \\ \hline \end{array}\)
