Answer
$\lim\limits_{w \to 2} \frac{f'(w)-f'(2)}{w-2} =3584$
Work Step by Step
We notice that the expression $\lim\limits_{w \to 2} \frac{f'(w)-f'(2)}{w-2}$ is the limit definition for $f''(2)$.
We then calculate the second derivative of $f(x)$ using the power rule:
$f(x) = x^8-2x+3$
$f'(x) = 8x^7-2$
$f''(x) = 56x^6$
We then evaluate the second derivative at $x=2$:
$\lim\limits_{w \to 2} \frac{f'(w)-f'(2)}{w-2} = f''(2) = 56*(2^6) = 3584$
