Final answer:
To find the limit as (h→0) of [(f(x+h) - f(x)) / h], substitute the given function f(x) = x³ - 5 into the expression. Simplify and cancel out the terms involving h. The limit is 3x².
Step-by-step explanation:
To find the limit as (h→0) of [(f(x+h) - f(x)) / h], we can substitute the given function f(x) = x³ - 5 into the expression.
[(f(x+h) - f(x)) / h] = [(x+h)³ - 5 - (x³ - 5)] / h = [(x³ + 3x²h + 3xh² + h³ - 5 - x³ + 5)] / h = [3x²h + 3xh² + h³] / h = 3x² + 3xh + h².
Now, as h approaches 0, all the h terms will become 0. Therefore, the limit of [(f(x+h) - f(x)) / h] as (h→0) is equal to 3x².