Question

The limit represents f '(c) for a function f(x) and a number c. find f(x) and c. lim δx→0 [8 − 6(1 δx)] − 2 δx. A) f(x)=8−6x, c=1

B) f(x)=6−8x, c=2

C) f(x)=8−6x, c=−1

D) f(x)=6−8x, c=−2

Answer 1

The function f(x) is 8 - 6x, and the number c is 1. This is determined by simplifying the given limit expression and identifying the resemblance with the derivative of the function at x equals 1.

Step-by-step explanation:

To find the function f(x) and the number c from a given limit that represents f'(c), consider the limit expression given:

\(lim_{Δx \to 0} [8 - 6(1 + Δx)] - 2 Δx\)

Firstly, expand the expression within the square brackets:

\(8 - 6(1 + Δx) = 8 - 6 - 6Δx = 2 - 6Δx\)

Now, substitute this back into the original expression:

\(lim_{Δx \to 0} (2 - 6Δx) - 2 Δx\)

Combine like terms:

\(lim_{Δx \to 0} (2 - 8Δx)\)

As Δx approaches 0, the only term that does not vanish is the constant term 2. Therefore, the limit expression simplifies to 2. This tells us that f'(c) equals 2. The limit expression resembles the derivative of the function f(x) = 8 - 6x at the point where x equals 1 because the -6 coefficient is associated with the variable. Thus, f(x) = 8 - 6x and c = 1.

So, the correct answer is:

f(x) = 8 - 6x, c = 1