Asymptotes (College Board AP® Calculus BC): Study Guide

Horizontal asymptote

What is a horizontal asymptote?

• A horizontal asymptote is a horizontal line

  • that the graph of a function gets closer and closer to (but never touches or intersects)

  • as x becomes unbounded in the positive or negative direction

• On the following diagram, the horizontal asymptote is indicated by a dashed line

How can I identify horizontal asymptotes using limits?

• A function will have a horizontal asymptote if it has a finite limit at infinity

  • I.e. the line $y=c$ will be a horizontal asymptote for the graph of a function $f$ if

    • $\underset{x\to \infty }{\lim }f\left(x\right)=c$, or

    • $\underset{x\to -\infty }{\lim }f\left(x\right)=c$

• Horizontal asymptotes (if any) may therefore be determined by evaluating the limits at infinity

How can I identify horizontal asymptotes numerically?

• If a function has a horizontal asymptote, then its values will tend to a limit for large positive or negative values of $x$

• Consider, the function $f\left(x\right)=\frac{12+5e^{-{\displaystyle \frac{x}{1000}}}}{1+e^{-{\displaystyle \frac{x}{1000}}}}$

• Consider the values for large positive $x$

• The table suggests that $\underset{x\to \infty }{\lim }f\left(x\right)=12$

  • Note $f\left(100000\right)\ne 12$

  • It starts as 11.999... with a large number of 9s

  • Most calculators cannot differentiate between this value and 12

• Consider the values for large negative $x$

• The table suggests that $\underset{x\to -\infty }{\lim }f\left(x\right)=5$

  • Note $f\left(-100000\right)\ne 5$

  • It starts as 5.000... with a large number of 0s

  • Most calculators cannot differentiate between this value and 5

Vertical asymptote

What is a vertical asymptote?

• A vertical asymptote is a vertical line

  • that the graph of a function gets closer and closer to (but never touches or intersects)

  • as x gets closer and closer to the x-value of the vertical line

• On the following diagram, the vertical asymptote is indicated by a dashed line

How can I identify vertical asymptotes using limits?

• A function will have a vertical asymptote at any x-value where the function becomes unbounded

  • I.e. the line $x=c$ will be a vertical asymptote for the graph of a function $f$ if

    • $\underset{x\to c^{-}}{\lim }f\left(x\right)=\pm \infty$, or

    • $\underset{x\to c^{+}}{\lim }f\left(x\right)=\pm \infty$

• Vertical asymptotes (if any) may therefore be determined by identifying points where the function becomes unbounded

  • Usually this will involve a function in the form of a quotient

    • at points where the denominator becomes zero

How can I identify vertical asymptotes numerically?

• If a function has a vertical asymptote, then its values tend to positive or negative infinity as the values of $x$ get closer to the vertical asymptote

• Consider, the function $f\left(x\right)=\csc x$

• Consider the values for values of $x$ just smaller than 0

• The table suggests that $\underset{x\to 0^{-}}{\lim }f\left(x\right)=-\infty$

  • The values of $f$ keep getting further from 0 as $x$ gets closer to 0

• Consider the values for values of $x$ just bigger than 0

• The table suggests that $\underset{x\to 0^{+}}{\lim }f\left(x\right)=\infty$

  • The values of $f$ keep getting further from 0 as $x$ gets closer to 0