Limit Integral

Let $f(x) = \frac{4x-2}{x^2+1}$ and let $F(x)$ be a anti-derivative of $f(x)$ such that $F'(x) = f(x)$

By The Fundamental Theorem of Calculus, we have $\begin{aligned} \lim_{h \to 0} \frac{1}{h} \int_{3-h}^{3+h} \frac{4x-2}{x^2+1} dx &= \lim_{h \to 0} \int_{3-h}^{3+h} f(x) dx \\ &= \lim_{h \to 0} \frac{F(3+h) - F(3-h)}{h} \\ &= \lim_{h \to 0} \frac{F(3+h) - F(3)}{h} + \lim_{i \to 0} \frac{F(3+i)-F(3)}{i} \text{ where } i = -h \\ &= F'(3) + F'(3) \\ &= f(3) + f(3) \\ &= 2 \times \frac{4 \times 3 - 2}{3^2 + 1} \\ &= 2 \ \end{aligned}$