Question. Find the limit of the following $$\lim_{n\to\infty}\int^{\infty}_0n\ln\Big(1+\frac{e^{-x}}{n}\Big).$$
Now here's my attempt
Define $$f_n(x)=n\ln\Big(1+\frac{e^{-x}}{n}\Big)=\ln\Big(1+\frac{e^{-x}}{n}\Big)^n\Rightarrow e^{f_n(x)}=\Big(1+\frac{e^{-x}}{n}\Big)^n.$$ But now I'm finding it hard to find a function that dominates $f_n(x)$ because I feel that using Lebesgue's dominated convergence theorem is the way to go here.
Also as an aside, do any of you know any resources that has similar questions like this as I feel as though I need more practice on these sorts of problems.
Thanks in advance!
$$L=\lim_{n\rightarrow \infty} \int _{0}^{\infty} n \ln \left( 1+\frac{e^{-z}}{n}\right) dx =\lim_{n\rightarrow \infty}\int_{0}^{\infty} n \left(\frac{e^{-x}}{n}-\frac{e^{-2x}}{n^2}+...\right) dx =\int_{0}^{\infty} e^{-x} dx=1.$$