From the Wikipedia page, dirac delta function is described such as:
Description 1: "whose value is zero everywhere except at zero" or
if you scroll down a little bit more, you will find:
Description 2:
$ \delta(x)= \begin{cases} 0&x \neq 0\\ \infty&x=0 \end{cases} $
While reading this, it made me believe that such function(even though it's not a function), its value is only at origin and anywhere else, it's $0$. This is what the above says.
Then, It also shows one of the family of function that "successfully" mimics the behaviour:
$\displaystyle\delta_b(x) = \lim_{b \to 0} \frac{1}{|b| \sqrt{\pi}} e^{-(\frac{x}{b})^2}$
Question: note that the following function doesn't correctly obey the description 1 or either description 2. If you calculate $\delta_b(x)$ at any $x>0.0000001$, it's zero, but just use $x=0.00000001$(extra zero), and the limit returns $\infty$. So we just found $x$ that is not $0$, but for which $\delta$ is not $0$. Could you explain which description is correct or what's going on ? I'm not good at math, so would appreciate the clear/easy explanation.
The meaning of the Dirac delta is not the same as what you are used to with "standard functions" (the Dirac delta is a distribution in fact), in fact the definition $$\delta(x)=\begin{cases}\infty&x= 0\\0&x\neq 0\end{cases}$$ it is not very useful for application purposes.
The most useful property of the Dirac delta is the fact that: $$\int_{-\infty}^{\infty}\delta(x)\mathrm{d}x=1$$ However, the meaning of the delta should not be sought in the sense of the functions where you said (for example) $f(5)=\pi$ but it must be found in relation to another function (generally called "test function")
The delta property comes into play when you do operations like this
$$\int_{-\infty}^{\infty}\delta(x-x_0)f(x)\mathrm{d}x=f(x_0)$$
From here you see that the use of delta is to sample functions when doing integrals (you don't evaluate the $\delta$ at a point like in functions but you evaluate other functions by means of integrals)
The fact that it is presented to you as the limit of $$\delta_b(x) = \lim_{b \to 0} \frac{1}{|b| \sqrt{\pi}} e^{-(\frac{x}{b})^2}$$ is just a construction to make you see that the function tends to $0$ everywhere except $0$ and in parallel you have that the area subtended by the function always remains $1$.