[tex]\qquad \boxed{x = \sqrt{19} + \dfrac{91}{x}}[/tex],
segue, por um processo de substituição em cascata, que:
[tex]\qquad x = \sqrt{19} + \dfrac{91}{\color{red}x}=\sqrt{19} + \dfrac{91}{\color{red}\sqrt{19}+\dfrac{91}{x}}[/tex]
[tex]\qquad \quad =\sqrt{19} + \dfrac{91}{\sqrt{19}+\dfrac{91}{\color{blue}x}}= \sqrt{19} + \dfrac{91}{\sqrt{19}+\dfrac{91}{\color{blue}\sqrt{19}+\dfrac{91}{x}}}[/tex]
[tex]\qquad \quad = \sqrt{19} + \dfrac{91}{\sqrt{19}+\dfrac{91}{\sqrt{19}+\dfrac{91}{\color{green}x}}}= \sqrt{19} + \dfrac{91}{\sqrt{19}+\dfrac{91}{\sqrt{19}+\dfrac{91}{\color{green}\sqrt{19}+\dfrac{91}{x}}}}[/tex]
[tex]\qquad \quad = \sqrt{19} + \dfrac{91}{\sqrt{19}+\dfrac{91}{\sqrt{19}+\dfrac{91}{\sqrt{19}+\dfrac{91}{\color{#0080Ff}x}}}}= \sqrt{19} + \dfrac{91}{\sqrt{19}+\dfrac{91}{\sqrt{19}+\dfrac{91}{\sqrt{19}+\dfrac{91}{\color{#0080Ff}\sqrt{19}+\dfrac{91}{x}}}}}[/tex]
(Em cada passagem, substituímos [tex]x[/tex] pela expressão [tex]\sqrt{19} + \dfrac{91}{x}[/tex].)
E assim, [tex]x = \boxed{ \sqrt{19} + \dfrac{91}{\sqrt{19}+\dfrac{91}{\sqrt{19}+\dfrac{91}{\sqrt{19}+\dfrac{91}{\sqrt{19}+\dfrac{91}{x}}}}}}[/tex].