Homework Help: Kernel Function

1. Apr 12, 2009

latentcorpse

I have a problem with my notes that I can't understand.

They say:

For the kernel function $K_{\delta}(x)=\frac{1}{\sqrt{2 \pi \delta}} e^{-\frac{x^2}{2 \delta}}$ for $\delta>0$,

we have as $\delta \rightarrow 0+ , K_{\delta}(x)= \infty$ if $x=0$ and $K_{\delta}(x)= 0$ if $x \neq 0$.

therefore $\lim_{\delta \rightarrow 0} K_{\delta}(x)$ does not exist.

doesn't this contradict itself? it says the limit doesn't exist but in the line before it just said what the limit was?

2. Apr 12, 2009

Cyosis

The limit only exists if you exclude x=0, which you don't. It's not a contradiction because they show that for different values of x you get different limits so the limit delta->0 does not exist for all x.

3. Apr 12, 2009

latentcorpse

why does the limit not exist if we include this point though?

is it because the limit of something cant be infinity (surely not?)

or is it because the limit of something cant be two different things (although i though this would have been ok too)

or is it because the limit isn't continuous?

4. Apr 12, 2009

Hurkyl

Staff Emeritus
No, it's because the limit doesn't exist. No function satisfies the definition of "limit of K_delta(x) as delta approahces 0".

5. Apr 12, 2009

Cyosis

They don't put any restrictions on x, which causes the function to have two different limits. For it to exist it should have the same limit for all possible values of x.