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

Let $n\in \Bbb N$ and define $\displaystyle I_n:=\int \limits _{1385}^{2006}f(nx)\,\mathrm dx-621$.
It holds that  $\displaystyle I_n =\int \limits _{1385}^{2006}f(nx)-1 \,\mathrm dx$, therefore $\displaystyle |I_n|\leq \int \limits _{1385}^{2006} \left \vert f(nx)-1\right \vert \,\mathrm dx$.
Recall $\lim \limits_{x\to +\infty}f(x)=1 \iff (\forall \delta >0)(\exists \varepsilon >0)(\forall \overline x\in \Bbb R)(\overline x>\varepsilon \implies |f(\overline x)-1|<\delta)$.
Take $\delta >0$. There exists $\varepsilon >0$ such that for all $\overline x>\varepsilon$ it holds that $|f(\overline x)-1|<\delta$. In particular, for large enough $n$, if $x\ge 1385$, then $|f(nx)-1|<\delta$.
It follows that $\displaystyle |I_n|\leq \int \limits _{1385}^{2006}\delta \,\mathrm dx=621\delta$ and $\lim \limits _{n\to +\infty}\left( |I_n|\right)\leq \lim \limits _{n\to +\infty}\left(621\delta\right)=621\delta$.
Since $\delta$ was an arbitrary positive real number, it was proved that $\lim \limits _{n\to +\infty}\left( |I_n|\right)$ is a lower bound of $\{621\delta \colon\delta >0\}$, therefore $\lim \limits _{n\to +\infty}\left( |I_n|\right)\leq \inf \left(\{621\delta \colon \delta >0\} \right)=0$.
Finally $$\displaystyle 0=\lim \limits _{n\to +\infty}\left( |I_n|\right)=\lim \limits _{n\to +\infty}\left( I_n\right)=\lim \limits _{n\to +\infty}\left( \int \limits _{1385}^{2006}f(nx)\,\mathrm dx-621\right),$$ thus $\displaystyle \int \limits _{1385}^{2006}f(nx)\,\mathrm dx =621.$
