Mathematics

Problem

A function

p

is defined for

x\neq 0

p(x)=\frac{\sin(6x)\,(1-\cos 2x)}{x^3}.

Work through the following tasks. (a) Express

1-\cos 2x

in an alternate trigonometric form that involves

\sin^2x

. Show your derivation. (b) Rewrite

p(x)

so that every factor either equals

x

or has

x

in its denominator. Do not combine unlike terms. (c) Use your expression from part (b) to compute

\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)

near the origin.

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