JEE Main 2020 (Online) 8th January Morning ShiftQuestion 1
$\lim_{x\to0}\left(\frac{3x^2+2}{7x^2+2}\right)^{1/x^2}$ is equal to :
Limits & Continuity
Solution
Concept. Base $\to\frac22=1$, exponent $\to\infty$: $1^\infty$ form. Formulas used. (1) $\lim f^g=e^{\lim g(f-1)}$Step 1 - $f-1$. $\frac{3x^2+2}{7x^2+2}-1=\frac{-4x^2}{7x^2+2}$. Step 2 - Exponent (1). $$\frac{1}{x^2}\cdot\frac{-4x^2}{7x^2+2}=\frac{-4}{7x^2+2}\to-2$$ Step 3 - Answer. Limit $=e^{-2}=\frac{1}{e^2}$. Answer: D