Unit 2 · Topic 1.7
Rational Functions and End Behavior
Compare the growth of the numerator and denominator, then use leading terms and polynomial division to describe horizontal, slant, or polynomial asymptotic behavior.
1. A Rational Function Compares Two Polynomials
A rational function is a quotient of polynomial functions:
\[R(x)=\frac{P(x)}{Q(x)},\qquad Q(x)\ne0.\]
Its output measures the numerator's value relative to the denominator's value. End behavior asks which polynomial grows faster as \(x\to\infty\) and as \(x\to-\infty\).
Excluded domain values matter to the complete graph, but end behavior concerns inputs of increasingly large magnitude.
2. Reduce the Comparison to Leading Terms
Suppose \(P\) has leading term \(a_nx^n\) and \(Q\) has leading term \(b_mx^m\). For large \(|x|\),
\[\frac{P(x)}{Q(x)}\sim\frac{a_nx^n}{b_mx^m}=\frac{a_n}{b_m}x^{n-m}.\]
The degree difference \(n-m\) identifies which polynomial dominates. The quotient of leading coefficients controls the scale and sign.
3. Master Degree-Comparison Table
| Degree relationship | Dominant comparison | End behavior | Asymptote type |
|---|
\(n| Denominator grows faster | \(R(x)\to0\) | Horizontal: \(y=0\) | |
| \(n=m\) | Neither dominates | \(R(x)\to a_n/b_m\) | Horizontal: \(y=a_n/b_m\) |
| \(n=m+1\) | Numerator grows faster | Follows a linear quotient | Slant line from division |
| \(n>m+1\) | Numerator grows faster | Follows a higher-degree quotient | Polynomial curve from division |
The degree comparison chooses the type. Polynomial division determines the complete equation whenever the numerator degree is greater.
4. Denominator Dominates: Horizontal Asymptote \(y=0\)
For
\[A(x)=\frac{5x-2}{2x^3+x+7},\]
the numerator has degree 1 and the denominator has degree 3. The leading-term quotient is
\[\frac{5x}{2x^3}=\frac{5}{2x^2}\to0.\]
Therefore
\[\lim_{x\to-\infty}A(x)=0,\qquad\lim_{x\to\infty}A(x)=0.\]
The graph has horizontal asymptote \(y=0\).
5. Equal Degrees: Ratio of Leading Coefficients
For
\[B(x)=\frac{-6x^4+3x-1}{2x^4-5x^2+8},\]
the degrees are equal, so neither polynomial dominates. The leading-term quotient approaches
\[\frac{-6x^4}{2x^4}=-3.\]
Thus \(y=-3\) is the horizontal asymptote and
\[\lim_{x\to\pm\infty}B(x)=-3.\]
6. Numerator One Degree Higher: Slant Asymptote
Consider the original example
\[C(x)=\frac{2x^3-5x+1}{x^2+4}.\]
The leading-term quotient \(2x^3/x^2=2x\) predicts linear end behavior. Polynomial division gives the exact decomposition:
\[C(x)=2x+\frac{-13x+1}{x^2+4}.\]
Since the remainder fraction approaches 0 in both directions, the graph approaches the line \(y=2x\).
\[\lim_{x\to\pm\infty}\big(C(x)-2x\big)=0.\]
The leading-term quotient predicts the slope, while division confirms the complete slant-asymptote equation.
7. Numerator More Than One Degree Higher
A rational function can approach a quadratic, cubic, or other polynomial curve. For
\[D(x)=\frac{x^4+2x^2+3}{x^2+1},\]
division gives
\[D(x)=x^2+1+\frac{2}{x^2+1}.\]
The remainder fraction approaches 0, so the polynomial asymptote is \(y=x^2+1\).
\[\lim_{x\to\pm\infty}\big(D(x)-(x^2+1)\big)=0.\]
Because the asymptote itself rises on both ends, \(D(x)\to\infty\) as \(x\to\pm\infty\).
8. Why Polynomial Division Works
When \(\deg P\ge\deg Q\), polynomial division writes
\[\frac{P(x)}{Q(x)}=S(x)+\frac{T(x)}{Q(x)},\qquad\deg T<\deg Q.\]
The denominator of the remainder fraction has higher degree than its numerator, so that fraction approaches 0. The graph of the rational function therefore gets arbitrarily close to the quotient polynomial \(S(x)\).
Use every term of \(S(x)\). Keeping only its leading term describes a broad tail direction but can give an incorrect asymptote equation.
9. What a Horizontal Asymptote Means
If \(y=b\) is a horizontal asymptote, outputs can be made and kept arbitrarily close to \(b\) by choosing inputs with sufficiently large positive or negative magnitude.
\[\lim_{x\to\infty}R(x)=b\quad\text{or}\quad\lim_{x\to-\infty}R(x)=b.\]
The two limits can be considered separately. A horizontal asymptote describes far-end behavior, not an excluded output value.
10. A Graph May Cross an End-Behavior Asymptote
Consider
\[E(x)=\frac{x^2+x}{x^2+1}.\]
Equal degrees give the horizontal asymptote \(y=1\). To check whether the graph crosses it, solve
\[\frac{x^2+x}{x^2+1}=1\quad\Longrightarrow\quad x=1.\]
The graph crosses its horizontal asymptote at \((1,1)\). Horizontal, slant, and polynomial asymptotes describe the tails; they are not barriers like vertical asymptotes.
11. Approach from Above or Below
After finding a horizontal asymptote \(y=b\), analyze the sign of \(R(x)-b\).
\[F(x)=\frac{3x^2+2x}{x^2+1},\qquad F(x)-3=\frac{2x-3}{x^2+1}.\]
| Direction | Sign of \(F(x)-3\) | Graph behavior |
|---|
| \(x\to-\infty\) | Negative | Approaches \(y=3\) from below |
| \(x\to\infty\) | Positive | Approaches \(y=3\) from above |
The asymptote value is the same in both directions even though the sides of approach differ.
12. Numerical Evidence for an Asymptote
For \(E(x)=(x^2+x)/(x^2+1)\), compare outputs with the horizontal asymptote \(y=1\).
| \(x\) | \(-100\) | \(-10\) | \(10\) | \(100\) |
|---|
| \(E(x)\) | \(0.9899\) | \(0.8911\) | \(1.0891\) | \(1.0099\) |
|---|
| \(E(x)-1\) | \(-0.0101\) | \(-0.1089\) | \(0.0891\) | \(0.0099\) |
|---|
The differences move toward 0 as input magnitude grows. Tables provide evidence, while degree comparison explains why the pattern continues.
13. Technology Workflow
- Identify numerator and denominator degrees and leading terms.
- Use the degree comparison to predict the asymptote type.
- If the numerator dominates, perform polynomial division.
- Graph both the rational function and the proposed asymptote.
- Expand the viewing window and check whether their vertical difference approaches 0.
- Use a table with large positive and negative inputs for numerical support.
- Write the final limit statement or difference limit.
14. Common Errors and AP Reasoning
- Setting every horizontal asymptote equal to \(y=0\).
- Using the ratio of leading coefficients when the degrees are unequal.
- Calling the leading-term quotient the exact slant asymptote without completing division.
- Discarding lower quotient terms from a polynomial asymptote.
- Assuming a graph cannot cross a horizontal or slant asymptote.
- Confusing end behavior as \(x\to\pm\infty\) with local behavior near a denominator zero.
- Describing only one end of the graph.
Complete response pattern: compare degrees, show the leading-term quotient or division, name the asymptote, and state what happens as \(x\to-\infty\) and \(x\to\infty\).