A rational function is written as a quotient of two polynomial functions: r(x) = N(x) ÷ D(x). Lesson 1.6 asked what a polynomial does at the far ends of its graph. Now we ask the same question about a quotient, and the answer depends on how the two polynomials compare.
Here is the key reading. At every input in its domain, r(x) gives a measure of the relative size of the numerator compared with the denominator. If r(x) = 50, the numerator is 50 times as big as the denominator. If r(x) = 0.02, the numerator is only a small slice of the denominator.
A rational function r(x) = N(x) ÷ D(x) is a quotient of two polynomials. At each input it tells you how big the numerator is compared with the denominator. Pick a function and an input and compare the two bars.
Lesson 1.6 showed that for inputs of large magnitude a polynomial is dominated by its leading term. A quotient of two polynomials is then a quotient of two leading terms. The polynomial with the greater degree has values that dominate the values of the rational function.
So the plan is always the same: write down the quotient of the leading terms, simplify it, and read the end behavior from the result. For r(x) = (x² + 1)/(x − 1), the leading terms are x² and x, and x² ÷ x = x. For huge inputs, r(x) behaves like x.
For inputs of large magnitude a polynomial acts like its leading term, so r(x) acts like the quotient of the leading terms. The last row shows r(x) ÷ (that quotient) settling toward 1.
| x | 10 | 100 | 1,000 | 10,000 |
|---|---|---|---|---|
| r(x) | 3.6458 | 3.0512 | 3.005 | 3.0005 |
| quotient of leading terms | 3 | 3 | 3 | 3 |
| r(x) ÷ quotient | 1.2153 | 1.0171 | 1.0017 | 1.0002 |
If the numerator has the greater degree, the polynomial in the numerator dominates the polynomial in the denominator for inputs of large magnitude. The quotient of the leading terms is then a nonconstant polynomial, and the original rational function has the end behavior of that polynomial. Everything you learned in Lesson 1.6 about even and odd degree and the sign of the leading coefficient now applies to the quotient.
If that polynomial is linear, the graph of the rational function has a slant asymptote parallel to the graph of the line. The curve follows the same direction as the line, but it can sit a little above or below it. Finding the exact equation of the slant asymptote comes later; here we only need the direction.
When the numerator has the greater degree, the quotient of the leading terms is a nonconstant polynomial, and r has the end behavior of that polynomial. When it is linear, a dashed line shows the direction of the slant asymptote.
| x | −1,000 | −100 | −10 | 10 | 100 | 1,000 |
|---|---|---|---|---|---|---|
| r(x) | −999 | −99 | −9 | 11 | 101 | 1,001 |
| quotient of leading terms | −1,000 | −100 | −10 | 10 | 100 | 1,000 |
If neither polynomial dominates the other, the two degrees are equal. Then the quotient of the leading terms is a constant, and that constant indicates the location of a horizontal asymptote of the graph.
| Function | Leading terms | Quotient | Horizontal asymptote |
|---|---|---|---|
| (3x² + 5x)/(x² − 4) | 3x² and x² | 3 | y = 3 |
| (5x − 2)/(2x + 1) | 5x and 2x | 5/2 | y = 5/2 |
| (2x² + 1)/(3 − x²) | 2x² and −x² | −2 | y = −2 |
The quotient of the leading coefficients is the whole answer. Watch the sign in the last row: the leading term of 3 − x² is −x², not 3.
Equal degrees mean neither polynomial wins. The quotient of the leading terms is a constant b, and the graph has a horizontal asymptote at y = b. The dashed line is y = b.
| x | −1,000 | −100 | −10 | 10 | 100 | 1,000 |
|---|---|---|---|---|---|---|
| r(x) | 2.995 | 2.9512 | 2.6042 | 3.6458 | 3.0512 | 3.005 |
| distance from b | 0.00499 | 0.0488 | 0.3958 | 0.6458 | 0.0512 | 0.00501 |
If the denominator has the greater degree, it dominates the numerator for inputs of large magnitude. The quotient of the leading terms is a rational function with a constant in the numerator and a nonconstant polynomial in the denominator, like 2/x or 5/x². As the inputs grow, the denominator grows and the whole fraction shrinks toward zero.
So the graph has a horizontal asymptote at y = 0.
When the denominator has the greater degree, the quotient of the leading terms is a constant over a nonconstant polynomial, and that shrinks toward zero as the inputs grow. The graph has a horizontal asymptote at y = 0.
| x | −1,000 | −100 | −10 | 10 | 100 | 1,000 |
|---|---|---|---|---|---|---|
| r(x) | −0.00199 | −0.0193 | −0.1287 | 0.2673 | 0.0207 | 0.00201 |
| quotient of leading terms | −0.002 | −0.02 | −0.2 | 0.2 | 0.02 | 0.002 |
When the graph of a rational function r has a horizontal asymptote at y = b, where b is a constant, the output values of r get arbitrarily close to b and stay arbitrarily close to b as the input values increase or decrease without bound. In limit notation:
Here the limit is a real number, not ∞. That is the big difference from polynomials: a nonconstant polynomial never levels off, but a rational function can. For the denominator-dominates case, b = 0. For the equal-degrees case, b is the quotient of the leading coefficients.
"Arbitrarily close" means you can force r(x) as near to b as you like, just by going far enough out. Choose how close you want to be (the tolerance), then see how far out the inputs must go, on both sides.
| distance from b | 101 | 102 | 103 | 104 | 105 | 106 |
|---|---|---|---|---|---|---|
| x = 10k | 0.2143 | 0.0224 | 0.00225 | 2.25×10⁻⁴ | 2.25×10⁻⁵ | 2.25×10⁻⁶ |
| x = −10k | 0.2368 | 0.0226 | 0.00225 | 2.25×10⁻⁴ | 2.25×10⁻⁵ | 2.25×10⁻⁶ |