Final answer:
To find f'(-2) using only the limit definition of derivatives, substitute the function into the equation and simplify the expression. Then evaluate the limit as h approaches 0.
Step-by-step explanation:
To find f'(−2) using only the limit definition of derivatives, we can start by writing the definition of the derivative:
f'(−2) = lim(h→0)(f(-2+h) - f(-2))/h
Next, substitute the function f into the equation and simplify the expression:
f'(−2) = lim(h→0)((2(-2+h)³ + 3(-2+h)² - 8(-2+h) + 5) - (2(-2)³ + 3(-2)² - 8(-2) + 5))/h
Finally, evaluate the limit as h approaches 0 to find the derivative of f at x = -2.