Undergraduate math competition problem: find $\lim \limits_{n \to \infty} \int \limits^{2006}_{1385}f(nx)\, \mathrm dx$ Suppose $f\colon [0, +\infty) \to \mathbb{R}$ is a continuous function and $\displaystyle \lim \limits_{x \to +\infty} f(x) = 1$. Find the following limit:
$$\large\displaystyle \lim \limits_{n \to \infty} \int \limits^{2006}_{1385}f(nx)\, \mathrm dx$$

From continuity there is a $\xi_ n \in (1385,2006)$ such that $\int _{1385}^{2006}f(n x) \mathrm{d}x = (2006-1385)f(n \xi _n)$ . 
Since $k_n= n \xi _n \to \infty$ we have
\begin{align}\lim _{n\to \infty}\int _{1385}^{2006}f(n x) \mathrm{d}x &=\lim _{n \to \infty} (2006-1385)f(n \xi _n)\\
&=(2006-1385)\lim _{n \to \infty} f(k_n)=(2006-1385)\end{align}
