Final answer:
To evaluate the given limit, we can use the binomial theorem to expand the expression and then simplify it. The final answer is 3x^2.
Step-by-step explanation:
To evaluate the limit lim (h → 0) [(x + h)^3 - x^3] / h, we can apply the binomial theorem and simplify the expression. The binomial theorem states that (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3. Using this, we can expand (x + h)^3 as x^3 + 3x^2h + 3xh^2 + h^3. Now, substitute the expanded expression back into the limit expression:
[(x + h)^3 - x^3] / h = (x^3 + 3x^2h + 3xh^2 + h^3 - x^3) / h = (3x^2h + 3xh^2 + h^3) / h = 3x^2 + 3xh + h^2. As h approaches 0, the term 3xh approaches 0 and the term h^2 approaches 0. Therefore, the limit is equal to 3x^2. So, the final answer is lim (h → 0) [(x + h)^3 - x^3] / h = 3x^2.