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 ℎ → 0 = (sin(𝜋(2+ℎ))−0)/ℎ

Answer 1

Recall that f'(x) = lim_h→0 [(f(x+h) - f(x)) / h]

So, if we let f(x) = sin(πx) and x = 2, then the given limit is f'(2).

Since f'(x) = πcos(πx), f'(2) = πcos(2π) = π