Understanding Limits of a Kernel Function

[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?

 

[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.

 

[latentcorpse]

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

is it because the limit of something can't be infinity (surely not?)

or is it because the limit of something can't be two different things (although i though this would have been ok too)

or is it because the limit isn't continuous?

 

[Hurkyl]

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

 

[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.