📝 题目
例 5 求证 $\displaystyle \lim \left( {\sqrt{{n}^{2} + n} - n}\right) = 1/2$ .
💡 答案与解析
证明 我们有
$$ \sqrt{{n}^{2} + n} - n = \sqrt{n}\left( {\sqrt{n + 1} - \sqrt{n}}\right) $$
$$ = \frac{\sqrt{n}}{\sqrt{n + 1} + \sqrt{n}}, $$
$$ \left| {\sqrt{{n}^{2} + n} - n - \frac{1}{2}}\right| = \left| {\frac{\sqrt{n}}{\sqrt{n + 1} + \sqrt{n}} - \frac{1}{2}}\right| $$
$$ = \frac{\sqrt{n + 1} - \sqrt{n}}{2\left( {\sqrt{n + 1} + \sqrt{n}}\right) } $$
$$ = \frac{1}{2{\left( \sqrt{n + 1} + \sqrt{n}\right) }^{2}} $$
$$ < \frac{1}{2{\left( 2\sqrt{n}\right) }^{2}} = \frac{1}{8n}. $$