# Limit of integral lead to proof of convergence to dirac delta

1. Feb 7, 2013

### lakmus

Hi,
I try to prove, that function
$f_n = \frac{\sin{nx}}{\pi x}$ converges to dirac delta distribution (in the meaning of distributions sure). On our course we postulated lemma, that guarantee us this if $f_n$
satisfy some conditions. So I need to show, that $\lim_{n\rightarrow \infty}\int_{a}^{b}f_n \mathrm{d}x$ is
$0$ when $0$ isn't in $[a,b]$ and
$1$ for $0$ in $(a,b)$ .

I never met with problem before, the integral isn't "clasical" function and I don't have clue, how could I even start. I tried do some limit proceses, but it didn't show any concrete value - just estimation . . . (for other function which I found on wiki was possible count the integral and the limit is after easy . . .)

Thaks for any help.

2. Feb 7, 2013

### mathman

Let y = nx and see what happens to the integral.

3. Feb 8, 2013

### lakmus

Well, I get
$\lim_{n\rightarrow \infty} \int_{\frac{a}{n}}^{\frac{b}{n}} \frac{\sin{y}}{\pi y}dy$
Right?
Actually I tried this before, but what now? Thanks and sorry, if it is obvious, but I can't see it . .
I tried do some limit processes, and if sign(a)=sign(b) and the function is there continues, that it
is integral from "zero" to "zero" which could be zero . . . ? But what about other case, can't imagine the way I could get the result 1 . . .

Last edited by a moderator: Feb 8, 2013
4. Feb 8, 2013

### mathman

You got the limits wrong. They should be axn and bxn, not a/n and b/n. The limit is just the infinite integral.

5. Feb 8, 2013

### lakmus

You can't imagine, how did you help me! (sure, shame on me - i did just really stupid mistake) But integral from minus to plus infinity I can calculate, so problem solved!
Thank you very much.