题目

求下列极限:-|||-(6) lim _(xarrow +infty )(x)^dfrac (3{2)}(sqrt (x+2)-2sqrt (x+1)+sqrt (x));

$\begin{aligned} &\text{求下列极限:} \\ &(6)\lim_{x\to+\infty}x^{\frac{3}{2}}(\sqrt{x+2}-2\sqrt{x+1}+\sqrt{x}); \end{aligned}$

题目解答

答案

解题如下：

解： $\begin{aligned} &\textbf{解}\lim_{x\to+\infty}x^{\frac{3}{2}}(\sqrt{x+2}-2\sqrt{x+1}+\sqrt{x}) \\ &=\lim_{x\to+\infty}\frac{x^{\frac32}(\sqrt{x+2}-2\sqrt{x+1}+\sqrt{x})(\sqrt{x+2}+2\sqrt{x+1}+\sqrt{x})}{\sqrt{x+2}+2\sqrt{x+1}+\sqrt{x}} \\ &=\lim_{x\to+\infty}\frac{2x^{\frac{3}{2}}(\sqrt{x^{2}+2x}-x-1)(\sqrt{x^{2}+2x}+x+1)}{(\sqrt{x+2}+2\sqrt{x+1}+\sqrt{x})(\sqrt{x^{2}+2x}+x+1)} \\ &=-2\lim_{x\to+\infty}\frac1{\left\{\sqrt{1+\frac2x}+2\sqrt{1+\frac1x}+1\right\}\left\{\sqrt{1+\frac2x}+1+\frac1x\right\}} \\ &=-\frac{1}{4}. \end{aligned}$