Also the derivative of a limit . . . a limit normally ends up being a constant number, the derivative of a constant . . . is 0.

Um . . . the formal definition for a limit that pertains to functions is not exactly that.

Limit Def:

Let f(x) be a function defined on an interval that contains x=a, except possibly at x=a. Then we say that,

if and only if for all there is a number such that

then we say:

The derivative on the other hand is very different.