The value of 4 + (1)/(5 + (1)/(4 + (1)/(5 + (1)/(4 + ∞)))) is:

Given $y = 4 + \dfrac{1}{5 + \mathop{\mathop{\dfrac{1}{4 + \dfrac{1}{5 + \dfrac{1}{4 + \ldots \ldots \infty}}}}_{⏟}}_{y}}$

$y = 4 + \dfrac{1}{\left(5 + \dfrac{1}{y}\right)}$

$⇒ y - 4 = \dfrac{y}{( 5y + 1 )}$

$⇒ 5{y}^{2} - 20y - 4 = 0$

$⇒ y = \dfrac{20 \pm \sqrt{{\left(- 20\right)}^{2} - 4 \cdot 5 \cdot \left(- 4\right)}}{2 \cdot 5}$

$⇒ y = \dfrac{20 \pm \sqrt{480}}{10}$

$∴ y = 2 - \sqrt{\dfrac{480}{100}}$$\to$rejected

So, $y = 2 + \sqrt{\dfrac{480}{100}} = 2 + \dfrac{2}{5}\sqrt{30}$

Correct with Option (A).