Evaluating $\lim_{x \to \infty}\frac{1}{x}\int_0^x|\sin(t)|dt$ I would appreciate some help with this problem:
Evaluate:
$$\lim_{x \to \infty}\frac{1}{x}\displaystyle\int_0^x|\sin(t)|dt$$

Note that $\vert\sin(t)\vert$ is non-negative, periodic with period $\pi$, and that 
$$\int_0^\pi\vert \sin(t)\vert dt=2.$$
Let $f(x)$ be the largest integer smaller than or equal to $x/\pi$. Then it holds that
$$\int_0^{f(x)\pi}\vert\sin(t)\vert dt\leq\int_0^x\vert\sin(t)\vert dt\leq\int_0^{[f(x)+1]\pi}\vert\sin(t)\vert dt.$$
This can be written as
$$2f(x)\leq\int_0^x\vert\sin(t)\vert dt\leq2[f(x)+1].$$
Dividing by $x$ and noting that $\lim_{x\to\infty}f(x)/x=1/\pi$ it follows that
$$\frac2\pi\leq\lim_{x\to+\infty}\frac1x\int_0^x\vert\sin(t)\vert dt\leq\frac2\pi.$$
