How to Interpret and Calculate Asymptotes of a Function

Asymptotes are lines that the graph approaches, but never meets. Asymptotes are imaginary lines that you want to draw as dashed lines, so it’s easy to see where they are while indicating that they aren’t part of the graph.

Theory

VerticalAsymptotes

The line $x=a$ is a vertical asymptote if $f\left(x\right)\to ±\infty$ when $x\to a$.

Vertical asymptotes occur when the denominator of a fraction is zero, because the function is undefined there.

Theory

Horizontalasymptotes

The line $y=c$ is a horizontal asymptote if $f\left(x\right)\to c$ when $x\to ±\infty$.

Note! There are also oblique (slanted) asymptotes, but they are not as common.

Example 1

You have the expression $f\left(x\right)=\frac{{x}^{2}+2x+1}{x-2}$. Find any vertical and horizontal asymptotes.

Vertical Asymptotes

To find the vertical asymptotes, you need to set the denominator equal to zero:

 $x-2=0⇔x=2$

Horizontal Asymptotes

You find the horizontal asymptotes by calculating the limit: $\begin{array}{llll}\hfill \underset{x\to \infty }{\mathrm{lim}}\frac{{x}^{2}+2x+1}{x-2}& =\underset{x\to \infty }{\mathrm{lim}}\genfrac{}{}{1.0pt}{}{\phantom{\rule{0.17em}{0ex}}\frac{{x}^{2}}{{x}^{2}}+\frac{2x}{{x}^{2}}+\frac{1}{{x}^{2}}\phantom{\rule{0.17em}{0ex}}}{\phantom{\rule{0.17em}{0ex}}\frac{x}{{x}^{2}}-\frac{2}{{x}^{2}}\phantom{\rule{0.17em}{0ex}}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\underset{x\to \infty }{\mathrm{lim}}\genfrac{}{}{1.0pt}{}{\phantom{\rule{0.17em}{0ex}}1+\frac{2}{x}+\frac{1}{{x}^{2}}\phantom{\rule{0.17em}{0ex}}}{\phantom{\rule{0.17em}{0ex}}\frac{1}{x}-\frac{2}{x}\phantom{\rule{0.17em}{0ex}}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{1+0+0}{0}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & ⇒\text{divergent}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Note! The word “divergent” in this context means that the limit does not exist.

The figure shows the graph of the function in Example 1. As you can see, there is a vertical asymptote at $x=2$ and no horizontal asymptote. There is an oblique asymptote.