\lim_{x\to \infty }\frac{\sqrt{x}}{\sqrt{x+\sqrt{x+\sqrt{x}}}}=\lim_{x\to \infty }{\frac{\sqrt{x}}{\sqrt{x\left(1+\sqrt{{\frac{1}{x}}+\sqrt{\frac{1}{x^3}}\right)}}=\lim_{x\to \infty }{\frac{\sqrt{x}}{\sqrt{x}\cdot \sqrt{1+\sqrt{{\frac{1}{x}}+\sqrt{\frac{1}{x^3}}}}}}=\lim_{x\to \infty }{\frac{1}{\sqrt{1+\sqrt{{\frac{1}{x}}+\sqrt{\frac{1}{x^3}}}}}}=1