Holes of Rational Functions (College Board AP® Precalculus): Revision Note

Holes of rational functions

What is a hole of a rational function?

• A hole is sometimes represented by an open circle on a graph

• For a rational function, it corresponds to an input value for which the function is not defined

  • Unlike at a vertical asymptote, however, the function does not display unbounded behavior on either side of a hole

Where do holes occur in rational functions?

• Let $r\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$ be a rational function

  • i.e. where $p\left(x\right)$ and $q\left(x\right)$ are both polynomial functions

• If $c$ is a real zero of both $p$ and $q$, i.e. if $p\left(c\right)=0$ and $q\left(c\right)=0$

  • and if the multiplicity of $c$ as a real zero of $p$ is greater than or equal to its multiplicity as a real zero of $q$

  • then the graph of $r$ has a hole at $\mathit{x}\mathbf{=}\mathit{c}$

    • E.g. $\frac{\left(x+4\right){\left(x-3\right)}^4}{\left(x+1\right){\left(x-3\right)}^4}$ and $\frac{\left(x+4\right){\left(x-3\right)}^7}{\left(x+1\right){\left(x-3\right)}^4}$ both have holes at $x=3$

      • because in both cases the numerator and denominator become zero when $x=3$

      • and the multiplicity of $x=3$ in the numerator (4 or 7) is greater than or equal to its multiplicity in the denominator (4)

    • Note that the expression $\frac{\left(x+4\right){\left(x-3\right)}^4}{\left(x+1\right){\left(x-3\right)}^4}$ can be simplified to give $\frac{\left(x+4\right)\cancel{{\left(x-3\right)}^4}}{\left(x+1\right)\cancel{{\left(x-3\right)}^4}}=\frac{x+4}{x+1}$

      • and the expression $\frac{\left(x+4\right){\left(x-3\right)}^7}{\left(x+1\right){\left(x-3\right)}^4}$ can be simplified to give $\frac{\left(x+4\right){\left(x-3\right)}^3\cancel{{\left(x-3\right)}^4}}{\left(x+1\right)\cancel{{\left(x-3\right)}^4}}=\frac{\left(x+4\right){\left(x-3\right)}^3}{x+1}$

  • If the multiplicity of $c$ as a real zero of $p$ is less than its multiplicity as a real zero of $q$, then there is a vertical asymptote at $x=c$

How do I express the behavior of rational functions near holes using limit notation?

• If a rational function $r$ has a hole at $x=c$

  • then for input values near the hole the output values of the function will converge towards a particular value

  • The value they converge to determines the $\mathit{y}$-coordinate of the hole

• E.g. consider the behavior of $\frac{\left(x+4\right){\left(x-3\right)}^4}{\left(x+1\right){\left(x-3\right)}^4}$ close to $x=3$

  • The output value gets closer and closer to 1.75 as $x$ gets closer and closer to 3

• If output values of $r$ become arbitrarily close to some number $L$ as input values become sufficiently close to $\mathit{c}$

  • then the hole occurs at the point $\left(c,L\right)$

  • and the corresponding limit notation is

$\underset{x\to c}{\lim }r\left(x\right)=L$

• $\underset{x\to c}{\lim }$ can be read as "the limit as $x$ approaches $c$"

How can I determine the limit of a rational function at a hole?

• One way to determine the limit at a point where a rational function has a hole

  • is to calculate output values of the function nearer and nearer to the hole

    • and see what value they seem to converge towards

  • For example, see the table of values for $\frac{\left(x+4\right){\left(x-3\right)}^4}{\left(x+1\right){\left(x-3\right)}^4}$ above

• However a simpler method can be to algebraically simplify the expression for the function

  • and then find the value of the simplified expression at the input value where the hole occurs

• E.g. you can see from the table above that $\underset{x\to 3}{\lim }\frac{\left(x+4\right){\left(x-3\right)}^4}{\left(x+1\right){\left(x-3\right)}^4}=1.75$

  • Instead, start by simplifying the expression

    • $\frac{\left(x+4\right)\cancel{{\left(x-3\right)}^4}}{\left(x+1\right)\cancel{{\left(x-3\right)}^4}}=\frac{x+4}{x+1}$

  • Substitute $x=3$ into the simplified expression

    • $\frac{3+4}{3+1}=\frac{7}{4}\left(=1.75\right)$

  • That can be a much quicker way to find the limit (especially on a non-calculator section of the exam)

    • unless a question specifies that you must use another method