$\mathop {\lim }\limits_{x \to +\infty }(\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x})=\mathop {\lim }\limits_{x \to +\infty }\frac{x+\sqrt{x+\sqrt{x}}-x}{\sqrt{x+\sqrt{x+\sqrt{x}}}+\sqrt{x}}$ (nhân và chia $\sqrt{x+\sqrt{x+\sqrt{x}}}+\sqrt{x}$) $=\mathop {\lim }\limits_{x \to +\infty }\frac{\sqrt{x+\sqrt{x}}}{\sqrt{x+\sqrt{x+\sqrt{x}}}+\sqrt{x}}$
$=\mathop {\lim }\limits_{x \to +\infty }\frac{\sqrt{1+\frac{1}{\sqrt{x}}}}{\sqrt{1+\sqrt{\frac{1}{x}+\frac{1}{x\sqrt{x}}}}+1}$ (chia cả tử và mẫu cho $\sqrt{x}$)
$=\frac{\sqrt{1+0}}{\sqrt{1+\sqrt{0+0}}+1}$
$=\frac{1}{2}$.