# Dirac delta function as the limit of a seqquence

1. Jan 19, 2010

### krishna mohan

Dirac delta function as the limit of a sequence

Hi..

If I have a sequence which in some limit tends to infinity for x=0 and goes to zero for x$$\neq$$0, then can I call the limit as a dirac delta function?

If not, what are the additional constraints to be satisfied?

Last edited: Jan 19, 2010
2. Jan 19, 2010

### krishna mohan

Let me refine that..

If the integral of a member of the sequence over the whole real line is one, along with the conditions stated in previous post, can the limit of the sequence be identified with a dirac delta?

Or are the conditions of the first post enough??

3. Jan 19, 2010

### Hurkyl

Staff Emeritus
Let fn be any sequence of positive functions that converges (in the sense of distributions) to the dirac delta.

Then 2fn does not converge to the dirac delta.

Also, I believe $\sqrt{f_n}$ converges to 0.

4. Jan 19, 2010

### Hurkyl

Staff Emeritus
Also,
$$\lim_{n \rightarrow +\infty} g_n(x) = 0$$​
does not imply
$$\lim_{n \rightarrow +\infty} g_n = 0$$​
(the second limit in the sense of distributions)

In fact, I'm pretty sure you can find a sequence that satisfies the first equation whose limit is $\delta(x-1)$.