📝 题目
例 10 求 $\displaystyle \lim {n}^{2}\left( {1 - n\sin \frac{1}{n}}\right)$ .
💡 答案与解析
解 我们有
$$ \lim {n}^{2}\left( {1 - n\sin \frac{1}{n}}\right) = \lim {n}^{3}\left( {\frac{1}{n} - \sin \frac{1}{n}}\right) $$
$$ = \lim {n}^{3}\left\{ {\frac{1}{n} - \left\lbrack {\frac{1}{n} - \frac{1}{6{n}^{3}} + o\left( \frac{1}{{n}^{4}}\right) }\right\rbrack }\right\} $$
$$ = \lim \left\lbrack {\frac{1}{6} + o\left( \frac{1}{n}\right) }\right\rbrack = \frac{1}{6}. $$