Question

\(\lim\limits_{x\rightarrow+\infty}\left(\dfrac{x^3}{3x^2-4}-\dfrac{x^2}{3x+2}\right)\)

Answer 1

\(\lim\limits_{x\rightarrow+\infty}\left(\dfrac{x^3}{3x^2-4}-\dfrac{x^2}{3x+2}\right)\)

\(=\lim\limits_{x\rightarrow+\infty}\dfrac{x^3\left(3x+2\right)-x^2\left(3x^2-4\right)}{\left(3x^2-4\right)\left(3x+2\right)}\)

\(=\lim\limits_{x\rightarrow+\infty}\dfrac{3x^4+2x^3-3x^4+4x^2}{\left(3x^2-4\right)\left(3x+2\right)}\)

\(=\lim\limits_{x\rightarrow+\infty}\dfrac{2x^3+4x^2}{\left(3x^2-4\right)\left(3x+2\right)}\)

\(=\lim\limits_{x\rightarrow+\infty}\dfrac{2+\dfrac{4}{x}}{\left(3-\dfrac{4}{x^2}\right)\left(3+\dfrac{2}{x}\right)}\)

\(=\dfrac{2+0}{\left(3-0\right)\left(3+0\right)}=\dfrac{2}{9}\)