Consider the function f(x) = x3 − 4x2 + 2. calculate the limit of the difference quotient at x0 = 3 for f(x).

Given: f(x) = x³ - 4x² + 2

Note that
f(x+h) - f(x) = (x+h)³ - 4(x+h)² + 2 - (x³ - 4x² + 2)
                  = x³ + 3x²h + 3xh² + h³ - 4(x² + 2xh + h²) + 2 - x³ + 4x² - 2
                  = 3x²h + 3xh² + h³ - 8xh - 4h²

Therefore
[f(x+h) - f(x)]/h = 3x² +3xh + h² - 8x - 4h


That is 
f'(x) = 3x² - 8x
f'(3) = 3(3²) - 8(3) = 3

3