📝 题目
例 30 若 $f\left( x\right)$ 是连续的以 $T$ 为周期的周期函数,求证:
$$ \mathop{\lim }\limits_{{x \rightarrow \infty }}\frac{1}{x}{\int }_{0}^{x}f\left( t\right) \mathrm{d}t = \frac{1}{T}{\int }_{0}^{T}f\left( t\right) \mathrm{d}t. $$
💡 答案与解析
证 令 $\varphi \left( x\right) = {\int }_{0}^{x}f\left( t\right) \mathrm{d}t - \frac{x}{T}{\int }_{0}^{T}f\left( t\right) \mathrm{d}t$ ,只要证明 $\mathop{\lim }\limits_{{x \rightarrow \infty }}\frac{\varphi \left( x\right) }{x} = 0$ . 由