# Kernel Function

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?

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Cyosis
Homework Helper
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.

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?

Hurkyl
Staff Emeritus