Problem: Let $f: \mathbb{R} \to \mathbb{R} $ be continuous. Let $t\in \mathbb{R} $. Evaluate the following limit: \[ \lim_{h \to 0} \frac{1}{h}\int _{t-h}^{t+h} f(s)\mathrm{d} s. \]
Solution: Note that \[ \lim_{h \to 0} \frac{1}{h} \int _{t -h} ^ {t + h} f(s)~\mathrm{d}s \] is of the form $\frac{0}{0}$, so we can use the L'Hosptial's rule to evaluate the limit. \begin{align*} \lim_{h \to 0} \frac{\int _{t-h}^{t+h} f(s)~\mathrm{d} s}{h} & = \lim_{h \to 0} \frac{f(t + h) + f(t - h)}{1} \\ & = f(t) + f(t) = 2f(t). \end{align*}