Limit and integral properties of a continuous function Let $f$ be a continuous function on $[0,\infty)$ such that $\displaystyle\lim_{x \to \infty}f(x)= c$.
Show that $\displaystyle\lim_{x \to \infty} \frac{1}{x}\int_0^x f(s)\;ds = c$.
I've tried splitting the integral into $\int_0^M+ \int_M^x$ but I don't really know where to go from there. Any help is appreciated.

If $c \neq 0$, then notice that $\int_0^x f(s)\;ds$ approaches $\pm\infty$ as $x \to \infty$. So your limit is an indeterminate form of type $\frac{\infty}{\infty}$.
