15.1: Asymptotes
- Page ID
- 181512
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| Title | Level of Approach | Type | Length |
|---|---|---|---|
| A Review of Rational Functions | College Algebra (Algebra 2.5) | Review | 16:30 |
| Identifying the Vertical Asymptotes of Rational Functions | College Algebra (Algebra 2.5) | Lecture | 7:44 |
| Identifying the Horizontal Asymptotes of Rational Functions | College Algebra (Algebra 2.5) | Lecture | 16:19 |
| Identifying the Slant Asymptotes of Rational Functions | College Algebra (Algebra 2.5) | Lecture | 12:44 |
| Asymptotic Behavior of Rational Functions | Precalculus (Algebra 3) | Lecture | 56:49 |
Definitions and Theorems
A vertical asymptote of a graph is a vertical line \(x = a\) where the graph tends toward positive or negative infinity as the inputs approach \(a\) from either the left or the right. We write\[ \begin{array}{rcll}
f(x) \to \infty & \text{ as } & x \to a^- & \text{ if the graph increases without bound as }x\text{ approaches }a\text{ from the left,} \\[6pt]
f(x) \to -\infty & \text{ as } & x \to a^- & \text{ if the graph decreases without bound as }x\text{ approaches }a\text{ from the left,} \\[6pt]
f(x) \to \infty & \text{ as } & x \to a^+ & \text{ if the graph increases without bound as }x\text{ approaches }a\text{ from the right,} \\[6pt]
& \text{ and } & & \\[6pt]
f(x) \to -\infty & \text{ as } & x \to a^+ & \text{ if the graph decreases without bound as }x\text{ approaches }a\text{ from the right.} \\[6pt]
\end{array} \nonumber \]
The end behavior of a graph is a description, usually as a function, of what the function values of the graph tend to approach as the inputs increase or decrease without bound.
A horizontal asymptote of a graph is a horizontal line \(y=b\) where the graph approaches the line as the inputs increase or decrease without bound. We write\[ \begin{array}{rcll}
f(x) \to b & \text{ as } & x \to \infty & \text{ if the graph approaches the line }y = b\text{ as }x\text{ increases without bound} \\[6pt]
& \text{ and } & & \\[6pt]
f(x) \to b & \text{ as } & x \to -\infty & \text{ if the graph approaches the line }y = b\text{ as }x\text{ decreases without bound.} \\[6pt]
\end{array} \nonumber \]
A single point where the graph of a function is not defined, indicated by an open circle on the graph, is called a removable discontinuity.
A removable discontinuity for the graph of a rational function occurs at \(x=a\) if
- \(a\) is a zero for a factor in the denominator that is common with a factor in the numerator, and
- after simplification, \( a \) is no longer a zero of the denominator.
- Proof
- The most elegant proof of this theorem occurs in Calculus.
Let \( f(x) = \frac{N(x)}{D(x)} \) be a rational function. Then
- \( f(x) \) has a horizontal asymptote of \( y = 0 \) if \( \deg(D(x)) > \deg(N(x)) \)
- \( f(x) \) has a horizontal asymptote of \( y = \frac{a}{b} \) if \( \deg(D(x)) = \deg(N(x)) \), where \( a \) is the lead coefficient of \( N(x) \) and \( b \) is the lead coefficient of \( D(x) \)
- \( f(x) \) has a slant or oblique asymptote if \( \deg(D(x)) < \deg(N(x)) \). The asymptote is a slant asymptote if \( \deg(N(x)) = \deg(D(x)) + 1 \); otherwise, it is an oblique asymptote. In either case, the equation of the asymptote is the quotient after dividing \( N(x) \) by \( D(x) \).