JEE Main 2025 — 23 January, Evening Shift — Question 20

$\lim _{x \rightarrow \infty} \frac{\left(2-\frac{3}{x}+\frac{5}{x^{2}}\right)\left(1-\frac{1}{3 x}\right)^{x / 2}}{\left(3+\frac{5}{x}+\frac{4}{x^{2}}\right)\left(1+\frac{2}{3 x}\right)^{x / 2}}$

$$\begin{aligned} & =\lim _{x \rightarrow \infty} \frac{2}{3} \cdot \frac{e^{\frac{x}{2}\left(1-\frac{1}{3 x}-1\right)}}{e^{\frac{x}{2}\left(1+\frac{2}{3 x}-1\right)}} \\& =\frac{2}{3} \cdot \mathrm{e}^{-\frac{1}{6}} \mathrm{e}^{1 / 3}=\frac{2}{3} \mathrm{e}^{-\frac{1}{2}} \end{aligned}$$