JEE Main 2013 (Offline)Question 1
$\lim_{x\to0}\frac{(1-\cos2x)(3+\cos x)}{x\tan4x}$ is equal to
Limits & Continuity
Solution
Concept. Use $1-\cos2x=2\sin^2x$ and $\frac{\tan4x}{x}\to4$. Formulas used. (1) $1-\cos2x=2\sin^2x$(2) $\lim_{x\to0}\frac{\sin x}{x}=1$, $\lim_{x\to0}\frac{\tan4x}{x}=4$Step 1 - Rewrite (1). $$2\Big(\frac{\sin x}{x}\Big)^2\cdot\frac{x}{\tan4x}\cdot(3+\cos x)$$ Step 2 - Limits (2). $2\cdot1\cdot\frac14\cdot4=2$. Answer: D