Limits, Continuity & Differentiability
Product of cos — Taylor Expansion
nta_pyq_2024_apr
Grade 12
Question:
The value of $\displaystyle\lim_{x\to0}2\left(\frac{1-\cos x\sqrt{\cos 2x}\sqrt[3]{\cos 3x}\cdots\sqrt[10]{\cos 10x}}{x^2}\right)$ is
Step-by-Step Solution
Key Concept: Use $\ln(1-u)\approx-u$ and $\cos(kx)\approx1-\frac{k^2x^2}{2}$ for small $x$. The product $\prod_{k=1}^{10}(\cos kx)^{1/k}\approx\exp\left(-\sum_{k=1}^{10}\frac{kx^2}{2}\right)$.
$\sum_{k=1}^{10}k=55$. Limit $=55$.
Correct Answer: 55