📝 题目
例 1 求下列极限:
(1) $\displaystyle{\mathop{\lim }\limits_{{x \rightarrow 0 + 0}}x\left\lbrack \frac{1}{x}\right\rbrack}$ ; (2) $\displaystyle{\mathop{\lim }\limits_{{x \rightarrow 0 - 0}}x\left\lbrack \frac{1}{x}\right\rbrack}$ .
💡 答案与解析
解 (1) $\left\lbrack \frac{1}{x}\right\rbrack \leq \frac{1}{x} < \left\lbrack \frac{1}{x}\right\rbrack + 1 \Rightarrow 1 - x < x\left\lbrack \frac{1}{x}\right\rbrack \leq 1\left( {\forall x > 0}\right)$
$$ \Rightarrow \mathop{\lim }\limits_{{x \rightarrow 0 + 0}}x\left\lbrack \frac{1}{x}\right\rbrack = 1\text{ . } $$
(2) $\left\lbrack \frac{1}{x}\right\rbrack \leq \frac{1}{x} < \left\lbrack \frac{1}{x}\right\rbrack + 1 \Rightarrow 1 \leq x\left\lbrack \frac{1}{x}\right\rbrack < 1 - x\left( {\forall x < 0}\right)$
$$ \Rightarrow \mathop{\lim }\limits_{{x \rightarrow 0 - 0}}x\left\lbrack \frac{1}{x}\right\rbrack = 1\text{ . } $$