Question

Calculate the following limit:

Answer 1

`\lim_(x\to\infty)\frac{\sqrt x}{\sqrt{x+√(x+\sqrt x)}}=\\ \lim_(x\to\infty)\frac{(\sqrt x)/(\sqrt x)}{\frac{\sqrt{x+√(x+\sqrt x)}}{\sqrt x}}=\\ \lim_(x\to\infty)\frac{1}{\sqrt{(x+√(x+\sqrt x))/(x)}}=\\ \lim_(x\to\infty)\frac{1}{\sqrt{1+(√(x+\sqrt x))/(x)}}=\\ \lim_(x\to\infty)\frac{1}{\sqrt{1+(√(x+\sqrt x))/(√(x^2))}}=\\`

`\lim_(x\to\infty)\frac{1}{\sqrt{1+\sqrt{(x+\sqrt x)/(x^2)}}}=\\\lim_(x\to\infty)\frac{1}{\sqrt{1+\sqrt{(1)/(x)+(\sqrt x)/(√(x^4))}}}=\\\lim_(x\to\infty)\frac{1}{\sqrt{1+\sqrt{(1)/(x)+\sqrt{(x)/(x^4)}}}}=\\ \lim_(x\to\infty)\frac{1}{\sqrt{1+\sqrt{(1)/(x)+\sqrt{(1)/(x^3)}}}}=\\ =\frac{1}{\sqrt{1+\sqrt{0+√(0)}}}=\\`

`=(1)/(√(1+0))=\\ =(1)/(√(1))=\\ =(1)/(1)=\\ 1`