Find limit of $\lim_{n \rightarrow \infty} \int_{-\infty}^\infty \cos(\frac{x^2+7n}{n})\cdot e^{-|x|} \, dx$

Question

I am trying to determine: $$\lim_{n \rightarrow \infty} \int_{-\infty}^\infty \cos(\frac{x^2+7n}{n})\cdot e^{-|x|} \, dx$$

I am thinking that the way to go is to use some theorem (maybe dominated convergence theorem) so that I can switch integral and sum: $$ \lim_{n \rightarrow \infty} \int_{-\infty}^\infty \cos(\frac{x^2+7n}{n})\cdot e^{-|x|} \, dx \leq \lim_{n \rightarrow \infty} \int_{-\infty}^\infty 1\cdot e^{-|x|} \, dx=1 \cdot 2$$

And with the constant function Lebesgue integrable we get: $$\int_{-\infty}^\infty \lim_{n \rightarrow \infty} \cos(x^2/n+7)\cdot e^{-|x|} \, dx=\cos(7)\cdot \int_{-\infty}^\infty e^{-|x|} \, dx=\cos(7)\cdot2$$

However I am not sure if I can set 2=w as the dominating function on the infity interval

Any hints or comments on error will be very appreciated

Answer 1

For $f_n(x)=\cos{(\frac{x^2+7n}{n})}e^{-|x|}$ we have that $|f_n(x)| \leq e^{-|x|},\forall n \in \Bbb{N}$ since $|\cos{(\frac{x^2+7n}{n})}| \leq 1$

So you can use the dominated convergence theorem(D.C.T).

The limit you found is correct. You just need the justification in the pink to use D.C.T.