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


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.