\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}.