lim_x→∞ [(x³ / (3x² - 4)) - (x² / (3x + 2))] =

\(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow \infty}\left(\frac{x^{3}}{3 x^{2}-4}-\frac{x^{2}}{3 x+2}\right) \\ \Rightarrow \lim _{x \rightarrow \infty} \frac{x^{3}(3 x+2)-x^{2}\left(3 x^{2}-4\right)}{\left(3 x^{2}-4\right)(3 x+2)} \\ \Rightarrow \lim _{x \rightarrow \infty} \frac{3 x^{4}+2 x^{3}-3 x^{4}+4 x^{2}}{\left(3 x^{2}-4\right)(3 x+2)} \\ \Rightarrow \lim _{x \rightarrow \infty} \frac{2 x^{3}+4 x^{2}}{\left(3 x^{2}-4\right)(3 x+2)} \\ \Rightarrow \lim _{x \rightarrow \infty} \frac{2 x^{2}(x+2)}{9 x^{3}+6 x^{2}-12 x-8} \\ \Rightarrow 2 \lim _{x \rightarrow \infty} \frac{1+\frac{2}{x}}{9+\frac{6}{x}-\frac{12}{x^{2}}-\frac{8}{x^{3}}} \\ \Rightarrow 2 \times \frac{1}{9}\\=\frac{2}{9} \end{array}\end{array} \)