What Distinguishes the Existence of lim_{x \rightarrow x_0} f'(x) from f'(x_0)?

[kittybobo1]

Homework Statement

I had a quick question concerning some definitions. What is the difference between lim_{x \rightarrow x_0} f'(x) to exist and for f'(x_0) to exist, definition wise?

Homework Equations

The Attempt at a Solution

 

[VeeEight]

If the function is continuous then the limit as x approaches y of f is equal to f evaluated at y. Continunity is not a required property of derivatives (there are examples to show this).

 

[HallsofIvy]

To take a very simple example, let f(x)= x² if x is not 1, f(1)= 2. For any x other than 1, f(x)= x² in some interval around 1 and so it's derivative is 2x. limit as x goes to 1 of f'(x) is 2. But since f(x) is not continuous at x= 1, it is not differentiable there.