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

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

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

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

\(=\frac{2}{3}\lim _{x\to \infty }{\left(\frac{\left(1-\frac{1}{3x}\right)}{\left(1+\frac{2}{3x}\right)}\right)}^{x/2}\)

\(=\frac{2}{3}\cdot \frac{{e}^{\frac{x}{2}\left(1-\frac{1}{3x}-1\right)}}{{e}^{\frac{x}{2}\left(1+\frac{2}{3x}-1\right)}}\)

\(=\frac{2}{3}\cdot \frac{{\mathrm{e}}^{-\frac{1}{6}}}{{\mathrm{e}}^{1/3}}=\frac{2}{3}{\mathrm{e}}^{-\frac{1}{2}}\)