📝 题目
例 31 设 $f\left( x\right) \in C\left\lbrack {a,b}\right\rbrack ,A < a < b < B$ . 求证:
$$ \mathop{\lim }\limits_{{h \rightarrow 0}}{\int }_{a}^{b}\frac{f\left( {x + h}\right) - f\left( x\right) }{h}\mathrm{\;d}x = f\left( b\right) - f\left( a\right) . $$
💡 答案与解析
证 原式 $= \mathop{\lim }\limits_{{h \rightarrow 0}}\frac{{\int }_{a}^{b}f\left( {x + h}\right) \mathrm{d}x - {\int }_{a}^{b}f\left( x\right) \mathrm{d}x}{h}$
$$ = \mathop{\lim }\limits_{{h \rightarrow 0}}\frac{{\int }_{a + h}^{b + h}f\left( x\right) \mathrm{d}x - {\int }_{a}^{b}f\left( x\right) \mathrm{d}x}{h} $$
$$ = \mathop{\lim }\limits_{{h \rightarrow 0}}\frac{{\int }_{b}^{b + h}f\left( x\right) \mathrm{d}x - {\int }_{a}^{a + h}f\left( x\right) \mathrm{d}x}{h} $$
$$ = \mathop{\lim }\limits_{{h \rightarrow 0}}\frac{{\int }_{b}^{b + h}f\left( x\right) \mathrm{d}x}{h} - \mathop{\lim }\limits_{{h \rightarrow 0}}\frac{{\int }_{a}^{a + h}f\left( x\right) \mathrm{d}x}{h} $$
$$ \frac{\xi \in \left( {b,b + h}\right) }{\eta \in \left( {a,a + h}\right) }\mathop{\lim }\limits_{{\xi \rightarrow b}}f\left( \xi \right) - \mathop{\lim }\limits_{{\eta \rightarrow a}}f\left( \eta \right) = f\left( b\right) - f\left( a\right) . $$