x lim(2x^2-3x+5)(3x-1)^(x)/(2)(3x^2+5x+4)(3x+2)^x is equal to :

Question

$\underset{x	o \infty}{lim}\frac{(2{x}^{2}-3x+5)(3x-1{)}^{\frac{x}{2}}}{(3{x}^{2}+5x+4)\sqrt{(3x+2{)}^{x}}}$ is equal to :

Accepted Answer

$\begin{gathered} \underset{x	o \infty}{lim}\frac{(2-\frac{3}{x}+\frac{5}{{x}^{2}}){(1-\frac{1}{3x})}^{x/2}}{(3+\frac{5}{x}+\frac{4}{{x}^{2}}){(1+\frac{2}{3x})}^{x/2}} \ =\underset{x	o \infty}{lim}\frac{2}{3}\cdot \frac{{e}^{\frac{x}{2}(1-\frac{1}{3x}-1)}}{{e}^{\frac{x}{2}(1+\frac{2}{3x}-1)}} \ =\frac{2}{3}\cdot \frac{{e}^{-\frac{1}{6}}}{{e}^{1/3}}=\frac{2}{3}{e}^{-\frac{1}{2}} \end{gathered}$

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