Example 5. Find the asymptotes of the curve $y=\frac{x^{3}}{1-x^{2}}$.
Solution. The graph of this function has two vertical asymptotes $x=-1, x=1$, since

$$
\lim _{x \rightarrow-1} \frac{x^{3}}{1-x^{2}}=\lim _{x \rightarrow 1} \frac{x^{3}}{1-x^{2}}=\infty
$$

By isolating the integer part of the function through direct division or using the following simple transformations, we get

$$
y=\frac{x^{3}}{1-x^{2}}=\frac{x^{3}-x+x}{1-x^{2}}=\frac{x\left(x^{2}-1\right)+x}{1-x^{2}}=-x+\frac{x}{1-x^{2}}
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_7ef2114a321cf223ba99g-130.jpg?height=501&width=502&top_left_y=70&top_left_x=57)

Fig. 3.15

![](https://cdn.mathpix.com/cropped/2024_05_22_7ef2114a321cf223ba99g-130.jpg?height=512&width=505&top_left_y=61&top_left_x=627)

Fig. 3.16

Comparing with formula (3.15), we conclude that the line $y=-x$ is an oblique asymptote (Fig. 3.16).

3 Remark. Vertical asymptotes of the curve should be sought where the denominator of the function is zero.