calculus - Precise definition of limit on edge of function

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 :

$0-\delta$