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:
$$\large\displaystyle \lim \limits_{n \to \infty} \int \limits^{2006}_{1385}f(nx)\, \mathrm dx$$

Using the substitution $t=nx$, we get $I_n =  \int^{2006}_{1385}f(nx)dx =  \frac{1}{n}\int_{1385n}^{2006n} f(t) dt$. Let $I=2006-1385$.
Now let $\epsilon>0$, and choose $L>0$ such that if $t\ge L$, then $-\frac{\epsilon}{I} < f(t)-1 < \frac{\epsilon}{I}$.
Now choose $N\ge \frac{L}{1385}$. Then if $n \ge N$ and $t \in [1385n,2006n]$, we have $-\frac{\epsilon}{I} < f(t)-1 < \frac{\epsilon}{I}$. Integrating over $[1385n,2006n]$ and dividing by $n$ gives
$$ -\epsilon  < I_n -I < \epsilon $$
It follows that $\lim_n I_n = I$.
