Example 4. Find the asymptotes of the curve $f(x)=\frac{x^{2}+1}{x}$.
Solution. Since

$$
\lim _{x \rightarrow 0} \frac{x^{2}+1}{x}=\infty
$$

the line $x=0$ is a vertical asymptote (see formula (3.12)).

Further,

$$
f(x)=\frac{x^{2}+1}{x}=x+\frac{1}{x}=x+\alpha(x)
$$

where $\alpha(x) \rightarrow 0$ as $x \rightarrow \pm \infty$.

Thus, the line $y=x$ (see formula (3.15)) will be an asymptote. The curve is shown in Fig. 3.15.