JEE Main 2023 (Online) 8th April Morning ShiftQuestion 1
$\lim_{x\to0}\left(\left(\frac{1-\cos^{2}(3x)}{\cos^{3}(4x)}\right)\left(\frac{\sin^{3}(4x)}{\left(\log_e(2x+1)\right)^{5}}\right)\right)$ is equal to
Limits & Continuity
Solution
Concept. Replace each factor by its simplest equivalent. Formulas used. (1) $\sin u\sim u$, $\ln(1+u)\sim u$Step 1. $1-\cos^23x=\sin^23x\sim9x^2$; $\cos^34x\to1$. Step 2. $\sin^34x\sim64x^3$; $(\ln(1+2x))^5\sim32x^5$. Step 3. $$\lim\frac{9x^2\cdot64x^3}{32x^5}=18$$ Answer: C