We have, \( \Large x=\sqrt{1+\sqrt{1+\sqrt{1+.....\infty}}} \)

=> \( \Large x=\sqrt{1+x} \)

=>\( \Large x^{2}=1+x => x^{2}-x-1=0 \)

=> \( \Large x=\frac{1\pm \sqrt{1+4}}{2}=\frac{1\pm \sqrt{5}}{2} \)

As x > 0 we take only \( \Large x=\frac{1+\sqrt{5}}{2} \)