The set of all point $x$ such that $|x-a| < \delta$ is called a $\delta$ neighborhood of the point $a$. The set of all points $x$ such that $0<|x-a|<\delta$ in which $x=a$ is excluded, is called a deleted $\delta$ neighborhood of $a$ or an open ball radius $\delta$ about $a$.
I don't understand this definition (and because of that also the definition of a limit point):
Is $\delta$ just a random positive integer?
What exactly is the use of a $\delta$ neighborhood, I don't see how it could be meaningful at all.