# Evaluate the limit integral

I am doing this exercises, but I am stuck. I have to evaluate this limit $$\lim_{n\to \infty}\int_{\mathbb{R}} f_n(x)dx$$ where $$f_n(x)=(1+\frac{2x}{n})^ne^{-x^2}$$.

I have seen that $$f_n(x)\to e^{2x-x^2}$$, so my idea is to use the dominated convergence theorem. The problem is that I am not able to find a function $$g(x)$$ such that $$|f_n(x)|\leq g(x)$$ $$\forall x\in \mathbb{R}$$ and $$g$$ is integrable.

Could you give me some advices on which $$g$$ to use? Thanks in advance!

• why can't you just use $g(x) = e^{2x-x^2}$? – mathworker21 Jul 14 at 12:11
• yes, you are right. – Alessandro Pecile Jul 14 at 12:32

$$|f_n(x)| \leq e^{2|x|-x^{2}}$$ which is integrable. [$$(1+\frac a n)^{n} \leq e^{a}$$ for all $$a >0$$].