📝 题目
例 23 设 $f\left( x\right)$ 在(0,1)内有定义,且
$$ \mathop{\lim }\limits_{{x \rightarrow 0}}f\left( x\right) = 0,\;\mathop{\lim }\limits_{{x \rightarrow 0}}\frac{f\left( x\right) - f\left( \frac{x}{2}\right) }{x} = 0. $$
求证: $\mathop{\lim }\limits_{{x \rightarrow 0}}\frac{f\left( x\right) }{x} = 0$ .
💡 答案与解析
证 因为 $\mathop{\lim }\limits_{{x \rightarrow 0}}\frac{f\left( x\right) - f\left( \frac{x}{2}\right) }{x} = 0$ ,所以对任意给定的 $\varepsilon > 0,\exists \delta >$ 0,使得当 $x \in \left( {0,\delta }\right)$ 时,
$$ \left| \frac{f\left( x\right) - f\left( \frac{x}{2}\right) }{x}\right| < \frac{\varepsilon }{2} \Rightarrow \left| {f\left( x\right) - f\left( \frac{x}{2}\right) }\right| < \frac{\varepsilon }{2}x \tag{4.6} $$
$\forall t \in \left( {0,\delta }\right)$ ,由 (4.6) 得
$$ \left| {f\left( t\right) }\right| = \left\lbrack {\left\lbrack {f\left( t\right) - f\left( \frac{t}{2}\right) }\right\rbrack + \left\lbrack {f\left( \frac{t}{2}\right) - f\left( \frac{t}{{2}^{2}}\right) }\right\rbrack }\right\rbrack $$
$$ + \cdots + \left\lbrack {f\left( \frac{t}{{2}^{n - 1}}\right) - f\left( \frac{t}{{2}^{n}}\right) }\right\rbrack + f\left( \frac{t}{{2}^{n}}\right) $$
$$ \leq \left| {f\left( t\right) - f\left( \frac{t}{2}\right) }\right| + \left| {f\left( \frac{t}{2}\right) - f\left( \frac{t}{{2}^{2}}\right) }\right| $$
$$ + \cdots + \left| {f\left( \frac{t}{{2}^{n - 1}}\right) - f\left( \frac{t}{{2}^{n}}\right) }\right| + \left| {f\left( \frac{t}{{2}^{n}}\right) }\right| $$
$$ < \varepsilon \left( {\frac{t}{2} + \frac{t}{{2}^{2}} + \cdots + \frac{t}{{2}^{n}}}\right) + \left| {f\left( \frac{t}{{2}^{n}}\right) }\right| . \tag{4.7} $$
因为 $\mathop{\lim }\limits_{{x \rightarrow 0}}f\left( x\right) = 0$ ,所以对 (4.7) 令 $\displaystyle{n \rightarrow + \infty}$ 取极限得到
$$ \left| {f\left( t\right) }\right| \leq {t\varepsilon } \Rightarrow \left| \frac{f\left( t\right) }{t}\right| < \varepsilon \;\left( {\forall t \in \left( {0,\delta }\right) }\right) . $$
从而 $\mathop{\lim }\limits_{{x \rightarrow 0}}\frac{f\left( x\right) }{x} = 0$ .
\subsubsection{七、连续概念及其应用}