Final answer:
Using L'Hôpital's Rule to evaluate the given limit, the answer is calculated as -1/8.
Step-by-step explanation:
The question is asking to evaluate the limit as x approaches 2 of the expression (x-2)/(f(2)-2f(x)), given that f(2) = 4 and f'(2) = 4. We can use L'Hôpital's Rule to find the limit of ratios of functions where the functions have the form 0/0 or ±∞/±∞ when evaluating the limit as the variable approaches the point of interest.
L'Hôpital's Rule states that if we have a limit in the form lim[x→c] (g(x)/h(x)) where both g(x) and h(x) approach 0 or both approach ±∞ as x approaches c, and the derivatives g'(x) and h'(x) exist near c with h'(x) ≠ 0, then the original limit can be found using the limit of the derivatives:
lim[x→c] (g(x)/h(x)) = lim[x→c] (g'(x)/h'(x))
Applying L'Hôpital's Rule to this problem, we take the derivatives of the numerator and denominator with respect to x, knowing that f'(2) = 4. So, the derivative of the numerator x-2 is 1 and the derivative of the denominator f(2)-2f(x) with respect to x is -2f'(x). Substituting x=2 into our derivatives, we find that:
lim[x→2] (1/(-2f'(2))) = 1/(-2*4) = -1/8