Help finding the limit of $\lim_{n \to \infty}\prod_{k=1}^{n}\left(1+\frac{1}{n}f\left(\frac{k}{n}\right)\right)$.

$f$ is continuous on $[0,1]$, hence bounded.Lets assume that $n$ is large, so that the factors $1+\frac{1}{n}f(\frac{k}{n})>0$ hence taking logarithms makes sense. Now, $$\log(1+x) = x+O(x^2)$$ in particular, if we let $x = \frac{1}{n} \max_{[0,1]}(f)$, we see that $$\left|\log\left(1+\frac{1}{n}f\left(\frac{k}{n}\right)\right)-\frac{1}{n}f\left(\frac{k}{n}\right)\right|\leq \frac{C_f}{n^2} $$ for some constant $C_f$ which depends only on $f$. It follows that $$ \sum_{k=1}^n\left|\log\left(1+\frac{1}{n}f\left(\frac{k}{n}\right)\right)-\frac{1}{n}f\left(\frac{k}{n}\right)\right|\leq \frac{C_f}{n}\to 0 \text{ as } n \to \infty$$ so the two series converge to the same limit.