In an exercise to solve problems using dominated convergence theorem, I came across this:
$$ \lim_{n\rightarrow \infty} \int_{0}^n \bigg(1-\frac{x}{n}\bigg)^n \log(2+\cos(x/n)) \ dx $$ My attempt: I rewrote the integral as follows $$ \int_{0}^{\infty} f_n\ g_n\ h_n \ dx $$ where $f_n(x)=\bigg(1-\frac{x}{n}\bigg)^n, \ g_n(x)=1_{(0,n)}(x), \ h_n(x)=\log(2+\cos(x/n))$. Thus $\lim_{n} f_n(x)g_n(x)h_n(x)=e^{-x}. 1_{(0,\infty)}.\log(3)$.
But I'm unable to find a dominator $F(x)$ such that $f_n(x)g_n(x)h_n(x) \leq F(x)$ and $F(.)$ being integrable.