I'll assume you mean to take a limit in some sense as over double sequences indexed by natural numbers $n, \nu$ such that $\frac{\nu}{n}$ is not an integer. For $f_n$ an $\frac{1}{n}$-periodic function such that $\max f_n=-\min f_n =n$, let $g_n$ be defined by $f_n(x)=ng_n(nx)$. Then $g_n$ is $1$-periodic and has max and min equal to 1 and -1. Let $r=\frac{\nu}{n}$. Then you require that $r$ is not an integer.
Then writing the integral of interest in terms of $g_n$ and $r$ and $n$, you'll be very tempted to change variables and integrate from 0 to n. Do so.
Then using the periodicity of $g_n$, you'll want to break the integral into a sum of $n$ integrals. Finally, switch the order of summation and integration, be happy that you can look up or re-derive the formula for geometric sums, and be shocked (or not, but I was pleasantly surprised).