Each of the following limits represents the derivative of a function 𝑓 at some value 𝑥 = 𝑎. State a formula for 𝑓 and the value of 𝑎.

Question

𝑙𝑖𝑚 𝑥 → 3 = ((𝑥+1)^3⁄2)−8)/(𝑥−3)

Answer 1

f' '(x) = 𝑙𝑖𝑚 𝑥 → a (f(x) - f(a))/(x-a) = 𝑙𝑖𝑚 𝑥 → 3 = ((𝑥+1)^3/2−8)/(𝑥−3)

Therefore a = 3, and (𝑥+1)^3/2 = 4^3/2 = 2³ = 8

So

f(x) = (𝑥+1)^3/2 and a = 3