Let's find the asymptotes of $\frac{2x^3+3}{x^3+2}$, for real $x$

The function is defined on $ℝ\backslash \{-\sqrt[3]{2}\}$: a vertical asymptote can only be at a finite bound of this domain

As $x$ approaches $-\sqrt[3]{2}$ from the left, the function tends to $\mathrm{+\infty }$; from the right, to $\mathrm{-\infty }$

The function tends to infinity: the line $x=-\sqrt[3]{2}$ is a vertical asymptote

As $x$ tends to -∞ or to +∞, the function tends to $2$: the line $y=2$ is a horizontal asymptote

▶Asymptotes:   $\{x=-\sqrt[3]{2},y=2\}$

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