\(\text { Evaluate } \lim _{x \rightarrow \infty}\left(\frac{2 x^2-3 x+10}{3 x^2+4 x-2}\right) \frac{(3 x-1)^{\frac{x}{2…

Ans. (3)

Sol.

\(\begin{aligned}
& \lim _{x \rightarrow \infty} \frac{x^2\left(2-\frac{3}{x}+\frac{10}{x^2}\right)}{x^2\left(3+\frac{4}{x}-\frac{2}{x^2}\right)} \cdot\left(\frac{3 x-1}{3 x+2}\right)^{x / 2} \\
& \lim _{x \rightarrow \infty} \frac{2}{3} \times e^{\left(\frac{-3}{3 x+2}\right)\left(\frac{x}{2}\right)} \\
& =\frac{2}{3} e^{-\frac{1}{2}}=\frac{2}{3 \sqrt{e}}
\end{aligned}\)