Mathematics

Problem

Let

k

be a real constant, and define

q_k(x)=\frac{\sin(kx)-k\sin x}{x^3}.

Respond to the prompts below. (a) Show that if

k\neq 1

, the numerator of

q_k(x)

has a factor of

(k-1)x

for all

x

. Provide complete algebraic justification. (b) Determine all real values of

k

for which

\displaystyle\lim_{x\to 0}q_k(x)

exists and is finite. Fully justify your reasoning. (c) For each value of

k

found in part (b), evaluate the limit. Present your work in a logically organized manner. (d) Discuss how the choice of

k

influences the cancellation of leading–order terms in the numerator, and describe the resulting impact on the magnitude of

q_k(x)

near

x=0

Intuitive explanation

The key is recognizing that this is a limit problem in disguise—you're looking at a ratio of small quantities as $x \to 0$ , which screams for Taylor series. Expand both $\sin(kx)$ and $\sin(x)$ around $x=0$ to see what cancels and what remains, then divide by $x^3$ to find the leading behavior. The value of $k$ determines which terms survive, so pay attention to how the coefficient $k$ appears in each expansion.

Intuitive explanation

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