A function is defined for by p(x)= (6x) ,(1- 2x) x^3

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

A function pp is defined for x≠0x\neq 0 by p(x)=sin⁡(6x) (1−cos⁡2x)x3.p(x)=\frac{\sin(6x)\,(1-\cos 2x)}{x^3}. Work through the following tasks. (a) Express 1−cos⁡2x1-\cos 2x in an alternate trigonometric form that involves sin⁡2x\sin^2x. Show your derivation. (b) Rewrite p(x)p(x) so that every factor either equals xx or has xx in its denominator. Do not combine unlike terms. (c) Use your expression from part (b) to compute lim⁡x→0p(x)\displaystyle\lim_{x\to 0}p(x). Supply clear justification for each transformation. (d) Explain how the order of the zero in the numerator compares to that in the denominator, and state what this implies about the behavior of p(x)p(x) near the origin.

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