I wanted to know if I am right with this problem.
I have to determine whether the limit for $n \rightarrow \infty$ exists and, if so, I have to calculate it.
The Integral is:
$J_n=\int_{[0,n]}(\frac{1}{n}(1+\frac{x}{n})e^{\frac{-x}{n}})dλ(x)$
I have here some problems because of the $n$ in $\int_{[0,n]}$.
Can I use the theorem of dominant convergence? If yes:
Since $f_n:=\frac{1}{n}\left(1+\frac{x}{n}\right)e^{\frac{-x}{n}}$ is measurable, converges pointwise to $0$, and $|f_n|\leq g(x):=1$ we can prove the existence of the limit.
For calculating the limit we can do:
$$\lim_{n \rightarrow \infty}\int_{[0,n]}\left(\frac{1}{n}(1+\frac{x}{n})e^{\frac{-x}{n}}\right)dλ(x)=\int_{[0,n]}\lim_{n \rightarrow \infty}\left(\frac{1}{n}\left(1+\frac{x}{n}\right)e^{\frac{-x}{n}}\right)dλ(x)=0$$
Am I right?