\(\lim\limits_{x\rightarrow+\infty}\dfrac{\left(5-3x\right)^9}{\left(2+x\right)^6}=\lim\limits_{x\rightarrow+\infty}\dfrac{x^6\left(\dfrac{5}{x}-3\right)^6\left(5-3x\right)^3}{x^6\left(\dfrac{2}{x}+1\right)^6}=\lim\limits_{x\rightarrow+\infty}\left(\dfrac{\dfrac{5}{x}-3}{\dfrac{2}{x}+1}\right)^6\left(\dfrac{5}{x}-3\right)^3x^3\)
Do \(\left\{{}\begin{matrix}\lim\limits_{x\rightarrow+\infty}\left(\dfrac{\dfrac{5}{x}-3}{\dfrac{2}{x}+1}\right)^6\left(\dfrac{5}{x}-3\right)^3=\left(-\dfrac{3}{1}\right)^6\left(-3\right)^3=-3^9< 0\\\lim\limits_{x\rightarrow+\infty}x^3=+\infty\end{matrix}\right.\)
\(\Rightarrow\lim\limits_{x\rightarrow+\infty}\left(\dfrac{\dfrac{5}{x}-3}{\dfrac{2}{x}+1}\right)^6\left(\dfrac{5}{x}-3\right)^3x^3=-\infty\)
