Question

Evaluate $\underset{x\to \infty }{lim}\frac{\sqrt{x}}{\sqrt{x+\sqrt{x+\sqrt{x}}}}$

Answer 1

Let $x=\frac{1}{t^2}$ as $x\to \infty \Rightarrow t\to 0$

$=\underset{t\to 0}{lim}\frac{1}{t\sqrt{\frac{1}{t^2}}+\sqrt{\frac{1}{t^2}+\frac{1}{t}}}=\frac{1}{t\sqrt{\frac{1}{t^2}+\frac{\sqrt{t+1}}{t}}}$

$=\frac{1}{\sqrt{1+t\sqrt{t+1}}}=1$.