Question

how to prove that the limit e->+0 of integral (-inf,+inf) of (f(x)*exp(-x^2/4e)/(2pi*sqrt(e))) equals f(0)

Answer 1

i have no idea, but i am going to try and write it
\[\large \lim_{\epsilon\to 0^+}\int_{-\infty}^{\infty}f(x)\frac{\exp(\frac{x^2}{4\epsilon})}{2\pi\sqrt{\epsilon}}=f(0)\]

Answer 2

Thank you, satellite73, the way I wrote is unreadable

Answer 3

oh, and f(x) has derivatives of every N, all of them continuous, and there is R, such as f(x)=0 for any x: abs(x)>R

Answer 4

can i change epsilon/e to t ?

\[\large \lim_{t\to 0^+}\int_{-\infty}^{\infty}f(x)\frac{\exp(\color{red}{-}\frac{x^2}{4t})}{2\pi\sqrt{t~}}=f(0) \]

Answer 5

of course

Answer 6

does dominated convergence thm apply here ?

Answer 7

well, we are not supposed to use it, as we did not have it in our lectures

Answer 8

I have just looked through it on wiki, I think I can use it

Answer 9

um, ganeshie8, I see you have received a medal, that probably means that your post is practically the answer, but I still don't get it. Could you please explain it to me? :)

Answer 10

Hey no, I am kind of stuck. Let me tag few @eliassaab @Alchemista

Answer 11

http://math.stackexchange.com/questions/55137/is-this-a-delta-function-and-delta-as-limit-of-gaussian
I have found the solution, question is closed