Precise definition of limit on edge of function

Question

Suppose we are to follow the definition of limit of epsilon delta.

How can we define a limit at the end of it interval?

for example

$f(x)=\sqrt{x}, x\in[0,\infty)$

$\lim_{x\to 0}f(x)$

do we say that the limit does not exist at $x\to 0$? since the very existential of $\delta$ would be impossible as the function would not be defined at :

$0-\delta$

Answer 1

For any function $f:D\subset\Bbb R\to \Bbb R$ and $x_0\in\bar D$, $$\lim_{x\to x_0}f(x)=L$$ means that for all $ε>0$ exists a $δ>0$ so that for all $x\in D$ whenever $|x-x_0|<δ$ then $|f(x)-L|< ε$.

This is the general definition that easily can be modified to higher dimensional spaces or general metric spaces.

One sided limits are a specialty of real functions, as with a higher dimensional domains there is no easily defined "side".