JEE Main 2025 (Online) 7th April Morning ShiftQuestion 1
$\lim_{x\to0^{+}}\frac{\tan\left(5(x)^{\frac{1}{3}}\right)\log_e\left(1+3x^{2}\right)}{\left(\tan^{-1}3\sqrt{x}\right)^{2}\left(e^{5(x)^{\frac{4}{3}}}-1\right)}$ is equal to
Limits & Continuity
Solution
As $x\to0^+$: $\tan(5x^{1/3})\sim5x^{1/3}$, $\ln(1+3x^2)\sim3x^2$, $(\tan^{-1}3\sqrt x)^2\sim9x$, $e^{5x^{4/3}}-1\sim5x^{4/3}$. Limit $=\frac{15x^{7/3}}{45x^{7/3}}=\frac13$. Option (B).