While attempting to find the derivative of x3 directly from the definition of the derivative, a student wrote the following three consecutive lines:
f′(x)=h→0limh(x+h)3−x3=h→0limh3x2h+3xh2+h3=h→0lim(3x2+3xh+h2)
(a) For each "=" sign above, explicitly name the limit or algebraic property that justifies the transition.
(b) Complete the derivation to obtain the final simplified value of f′(x).
(c) In no more than four sentences, explain why the step where h tends to 0 is crucial for the final answer.