Infinite Limits and Asymptotes

The limits of $f$, as " $x \rightarrow \infty$ " and " $x \rightarrow-\infty$," give us information about horizontal asymptotes of $f$.

Example 6: Find any horizontal asymptotes of $\mathrm{f}(\mathrm{x})=\frac{2 \mathrm{x}+\sin (\mathrm{x})}{\mathrm{x}}$.

Solution: $\lim \limits_{x \rightarrow \infty} \frac{2 x+\sin (x)}{x}=\lim _{x \rightarrow \infty} \frac{2 x}{x}+\frac{\sin (x)}{x}=2+0=2$ so the line $\mathbf{y}=\mathbf{2}$ is a horizontal asymptote of $\mathrm{f}$. The limit, as " $\mathrm{x} \rightarrow-\infty$, " is also $2$ so $\mathrm{y}=2$ is the only horizontal asymptote of $\mathrm{f}$. The graphs of $\mathrm{f}$ and $\mathrm{y}=2$ are given in Fig. 7. A function may or may not cross its asymptote.

Fig. 7