# How to evaluate $\lim_{h\to0}\frac{1}{h}\int_{-h}^{h}f(t)dt?$

Let $f\in\mathcal[-\pi,\pi].$ How to evaluate $$\lim_{h\to0}\frac{1}{h}\int_{-h}^{h}f(t)dt?$$

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Let $F(t)=\int f(t)dt$. Then, $$\lim_{h\to0}\frac{1}{h}\int_{-h}^{h}f(t)dt=2\lim_{h\to0}\dfrac{F(-h+2h)-F(-h)}{2h}=\lim_{h\to0}2f(-h)=2f(0)$$