Final answer:
The correct limit as h approaches 0 of the expression [f(4h) - f(4)] / h for f(x) = 3x² is -48. However, this answer is not listed in the provided options, indicating a possible error in the question or answer choices.
Step-by-step explanation:
The student has asked to evaluate the limit as h approaches zero of the expression [f(4h) - f(4)] / h where f(x) = 3x². To solve this, plug the values into the function:
- f(4h) = 3(4h)² = 3(16h²) = 48h²
- f(4) = 3(4²) = 3(16) = 48
Now, substitute these into the limit expression:
lim_{h \to 0} \frac{48h² - 48}{h}
We can simplify this expression by canceling out terms:
lim_{h \to 0} \frac{48h(h - 1)}{h}
lim_{h \to 0} 48(h - 1)
As h approaches 0, the expression within the parentheses (h - 1) approaches -1. Therefore, the limit evaluates to:
48(-1) = -48
However, none of the provided multiple choice answers (A: 12, B: 24, C: 48, D: 0) match this result. Thus, there might be a mistake in the question as posed, or in the list of provided answers.