Question

In lecture two the last question he answers he says

limx as x->x0 x-x0 = 0
Is there a theorem or treatment on this?
He doesn't explain it.
Also is limx->x0 f(x)-f(x0) = 0
is that the theorem of differentiability?

Answer 1

This is just a statement in algebraic form indicating that as x approaches a specific value of x, that x=that value.
and
that as the limit of x approaches that value that the function with respect to x will equal the function at that value.
Basically common sense...
x=x0 and f)x)=f(x0)

This is not a theorem of differentiability.

Answer 2

I have one complaint. x approaches a specific value of x, it's not that value, it's infinitely approaching it!

Answer 3

No....it is not infinitely approaching anything....
I went to the lecture in question....he is basically stating that a function is continuous at a given value of x0 if the limit as x approaches that value exists and if the function at that value exists....he then goes on to talk about the limit from the right and left being equal which is required for the limit to exist....

There is nothing more to it than this....