# Chapter 11: Asymptotes

### Horizontal Asymptotes

Recall,

﻿$y=b$﻿ is a HORIZONTAL ASYMPTOTE of ﻿$f(x)$﻿ if ﻿$\lim_{x \rightarrow \infty} f(x) =b$﻿ or ﻿$\lim_{x \rightarrow - \infty} f(x) =b$﻿

#### Example

Find the horizontal asymptotes of ﻿$y = \frac{x^2}{2x^2+1}$﻿

We find the limits at infinity

﻿$\lim_{x \rightarrow \infty} \frac{x^2}{ 2x^2 +1}$﻿

﻿$=$﻿ ﻿$\lim_{x \rightarrow \infty} \frac {x^2}{2x^2 +1 } (\frac{1/x^2 }{1/x^2})$﻿ ﻿$\rightarrow$﻿ Cancel ﻿$x^2$﻿

﻿$= \lim_{x \rightarrow \infty} \frac{1}{2 + \frac{1}{x^2}}$﻿

﻿$= \frac {1}{2+0 }$﻿

﻿$=\frac{1}{2}$﻿

Thus, ﻿$y= \frac{1}{2}$﻿ is a horizontal asymptote

﻿$*$﻿ ﻿$\frac {\infty}{\infty}$﻿ is tricky. It can equal any finite value

#### EXample

Find the horizontal asymptotes of ﻿$y= \frac{3x^3 +5x}{10x^3 -3}$﻿

• Cancel ﻿$x^3$﻿
• Visually, the graph flattens

#### EXample

Find the horizontal asymptote of ﻿$y=e^x$﻿

Limits to infinity

﻿$\lim_{x \rightarrow \infty} e^x =\infty$﻿ and ﻿$\lim_{x \rightarrow \infty} e^x =0$﻿

Observe that ﻿$e^ x \geq 0$﻿ #### Example

How does ﻿$y = \frac {x^3} {x^4+1 }$﻿ approach the line ﻿$y=0$﻿?

﻿$\lim_{ x \rightarrow \infty} \frac{x^3}{x^4+1}$﻿﻿$= \frac{+}{+}$﻿

Thus , ﻿$\lim_{ x \rightarrow \infty} \frac{x^3}{x^4+1} = 0^+$﻿

﻿$\lim_{x \rightarrow - \infty} \frac{x^3}{x^4+1}=\frac{-}{+}$﻿

Thus, ﻿$\lim_{x \rightarrow \infty} \frac {x^3}{x^4+1 }= 0^-$﻿

The graph crosses ﻿$y=0$﻿

This graph was made using a graphing calculator Why are there bumps?

#### Example

How does ﻿$y= \frac {x^5}{6x^5 -3 }$﻿ approaches ﻿$y = \frac{1}{6}$﻿?

﻿$\lim_{x \rightarrow \infty} \frac{x^5 }{6x ^5-3 } = \frac{+}{+ }$﻿

Thus, ﻿$\lim_{x \rightarrow \infty} f(x) = \frac {1}{6} ^+$﻿

﻿$\lim_{x \rightarrow \infty} \frac{ x^5}{6x^5 - 3} = \frac{-}{-}$﻿

Thus ﻿$\lim_{x \rightarrow - \infty} f(x) = \frac {1}{6} ^+$﻿

### Vertical Asymptotes

#### EXample

Where are the vertical asymptotes of ﻿$y= \frac{x^5}{6x^5-3}$﻿?

Find zeroes of denominator

﻿$y= \frac{x^5}{6x^5-3} = \frac {x^5}{6(x^5- \frac{1}{2})}$﻿

﻿$6(x^5-\frac{1}{2}) = 0 \Leftrightarrow x^5- \frac{1}{2} =0$﻿

﻿$\Leftrightarrow x^5 = \frac {1}{2} \Leftrightarrow x = \frac{1}{ \sqrt{2}} =C$﻿

Find limit to ﻿$x =\frac{1}{ \sqrt{2}}$﻿

﻿$\lim_{x \rightarrow c^+} \frac{ x^5 }{6x^5-3 }= \frac{+}{+} = + \infty$﻿ ﻿$x> \frac{1}{\sqrt{2}} \Leftrightarrow 6x^5-3 >0$﻿

﻿$\lim_{x \rightarrow c^-} \frac{ x^5 }{6x^5-3 }= \frac{+}{-} = - \infty$﻿ ﻿$x < \frac{1}{\sqrt{2}} \Leftrightarrow 6x^5-3 <0$﻿

Thus, ﻿$\lim_{x \rightarrow c^+} f(x) = \infty$﻿ and ﻿$\lim_{x \rightarrow c^-} f(x) = - \infty$﻿

#### Discuss

What is the slope of this graph near asymptotes?

• Flattens out near horizontal.
• Spike near vertical
• Changes sign ## Summary

### Horizontal Asymptotes

• ﻿$y =b \Leftrightarrow \lim_{x \rightarrow \infty} f(x) = b$﻿ or ﻿$\lim_{x \rightarrow - \infty} f(x)=b$﻿
• 0, 1, or 2 horizontal asymptotes
• The graph flattens out.
• Approach from above ﻿$(y=b^+)$﻿ or below ﻿$(y=b^-)$﻿

### Vertical Asymptotes

• ﻿$x=c \Leftrightarrow \lim_{x \rightarrow c^+} f(x) = \pm \infty or \lim_{x \rightarrow c^-} f(x) = \pm \infty$﻿
• ﻿$x=c \Leftrightarrow \lim_{x \rightarrow c^+} f(x) = \pm \infty$﻿ or ﻿$\lim_{x \rightarrow c^-} f(x) = \pm \infty$﻿
• Any number of vertical asymptotes
• The graph has a spike
• The graph may change sign

#### Example

Classify all asymptotes of ﻿$y = \frac{x^3}{5x^3-5x}$﻿

#### EXample

Classify all asymptotes of ﻿$y = \frac{ \sin(x)}{ x}$﻿ Given the asymptotes how do we fill in the rest of the graph?