MA005: Calculus I

If the limit of $\mathrm{f}(\mathrm{x})$ as " $\mathrm{x} \rightarrow \infty$ " or " $\mathrm{x} \rightarrow-\infty$ " is a constant $\mathrm{K}$, then the graph of $\mathrm{f}$ gets close to the horizontal line $\mathrm{y}=\mathrm{K}$, and we said that $\mathrm{y}=\mathrm{K}$ was a horizontal asymptote of $\mathrm{f}$. Some functions, however, approach other lines which are not horizontal.

Example 8: Find all asymptotes of $\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}^{2}+2 \mathrm{x}+1}{\mathrm{x}}=\mathrm{x}+2+\frac{1}{\mathrm{x}}$.

Solution: If $\mathrm{x}$ is a large positive number or a large negative number, then $\frac{1}{x}$ is very close to $0$, and the graph of $f(x)$ is very close to the line $\mathrm{y}=\mathrm{x}+2$ (Fig. 8). The line $\mathbf{y}=\mathbf{x}+\mathbf{2}$ is an asymptote of the graph of $\mathrm{f}$.

Fig. 8

If $\mathrm{x}$ is a large positive number, then $1 / \mathrm{x}$ is positive, and the graph of $\mathrm{f}$ is slightly above the graph of $\mathrm{y}=\mathrm{x}+2$. If $\mathrm{x}$ is a large negative number, then $1 / \mathrm{x}$ is negative, and the graph of $\mathrm{f}$ will be slightly below the graph of $\mathrm{y}=\mathrm{x}+2$. The $1 / \mathrm{x}$ piece of $\mathrm{f}$ never equals $0$ so the graph of $\mathrm{f}$ never crosses or touches the graph of the asymptote $\mathrm{y}=\mathrm{x}+2$.

The graph of $f$ also has a vertical asymptote at $x=0$ since $\lim \limits_{x \rightarrow 0^{+}} f(x)=\infty$ and $\lim \limits_{x \rightarrow 0^{-}} f(x)=-\infty$.

Practice 7: Find all asymptotes of $\mathrm{g}(\mathrm{x})=\frac{2 \mathrm{x}^{2}-\mathrm{x}-1}{\mathrm{x}+1}=2 \mathrm{x}-3+\frac{2}{\mathrm{x}+1}$.

Some functions even have nonlinear asymptotes, asymptotes which are not straight lines. The graphs of these functions approach some nonlinear function when the values of $x$ are arbitrarily large.

Example 9: Find all asymptotes of $f(x)=\frac{x^{4}+3 x^{3}+x^{2}+4 x+5}{x^{2}+1}=x^{2}+3 x+\frac{x+5}{x^{2}+1}$.

Solution: When $x$ is very large, positive or negative, then $\frac{x+5}{x^{2}+1}$ is very close to $0$, and the graph of $\mathrm{f}$ is very close to the graph of $\mathrm{g}(\mathrm{x})=\mathrm{x}^{2}+3 \mathrm{x}$. The function $\mathrm{g}(\mathrm{x})=\mathrm{x}^{2}+3 \mathrm{x}$ is a nonlinear asymptote of $\mathrm{f}$. The denominator of $\mathrm{f}$ is never $0$, and $\mathrm{f}$ has no vertical asymptotes.

Practice 8: Find all asymptotes of $f(x)=\frac{x^{3}+2 \sin (x)}{x}=x^{2}+\frac{2 \sin (x)}{x}$.

If $\mathrm{f}(\mathrm{x})$ can be written as a sum of two other functions, $\mathrm{f}(\mathrm{x})=\mathrm{g}(\mathrm{x})+\mathrm{r}(\mathrm{x})$, with $\lim \limits_{x \rightarrow \pm \infty} \mathrm{r}(\mathrm{x})=0$, then the graph of $\mathrm{f}$ is asymptotic to the graph of $\mathrm{g}$, and $\mathrm{g}$ is an asymptote of $\mathrm{f}$.