Find all asymptotes of y=(x³)/(2(x+1)²)

Find all asymptotes of $y=\dfrac{x^3}{2(x+1)^2}$.

Step 1: Vertical asymptote

At $x=-1$ the denominator vanishes while the numerator equals $-1\neq0$, so $y\to-\infty$ from both sides: \[x = -1. \tag{1}\]

Step 2: Oblique asymptote — slope

\[ k = \lim_{x\to\infty}\frac{x^2}{2(x+1)^2} = \frac12. \tag{2} \]

Step 3: Oblique asymptote — intercept

\[ b = \lim_{x\to\infty}\!\left(y-\frac x2\right) = \lim_{x\to\infty}\frac{x^3-x(x+1)^2}{2(x+1)^2} = \lim_{x\to\infty}\frac{-2x^2-x}{2(x+1)^2} = -1. \tag{3} \] using $x^3-x(x+1)^2=x^3-x^3-2x^2-x=-2x^2-x$.

Step 4: Verification

\[x=500\text{: } y-(x/2-1)\approx-0.00200\to0;\qquad x\to-1^{\pm}\text{: }y\to-\infty\ \checkmark\]