A horizontal asymptote is a horizontal line that the graph of a function approaches as the input (x) tends towards positive or negative infinity. It describes the end behavior of the function. For rational functions (a function that is a ratio of two polynomials), horizontal asymptotes are determined by comparing the degrees of the numerator and the denominator.
How to Calculate Horizontal Asymptotes for Rational Functions
Consider a rational function in the form:
f(x) = P(x) / Q(x)
where P(x) is the numerator polynomial and Q(x) is the denominator polynomial. Let n be the degree of P(x) and m be the degree of Q(x).
Case 1: n < m (Degree of numerator is less than degree of denominator)
The horizontal asymptote is the line y = 0.
Case 2: n = m (Degree of numerator equals degree of denominator)
The horizontal asymptote is the line y = a/b, where a is the leading coefficient of the numerator and b is the leading coefficient of the denominator.
Case 3: n > m (Degree of numerator is greater than degree of denominator)
There is no horizontal asymptote. The function will tend towards positive or negative infinity. (Note: There might be a slant or oblique asymptote in this case, but this calculator focuses only on horizontal ones).
Example Calculation:
Let's find the horizontal asymptote for the function:
f(x) = (3x^3 - 5x + 2) / (2x^3 + 7x^2 - 1)
Numerator coefficients: 3, 0, -5, 2 (representing 3x³ + 0x² – 5x + 2). The degree (n) is 3.
Denominator coefficients: 2, 7, 0, -1 (representing 2x³ + 7x² + 0x – 1). The degree (m) is 3.
Since the degrees are equal (n = m = 3), we are in Case 2. The leading coefficient of the numerator is 3, and the leading coefficient of the denominator is 2.
Therefore, the horizontal asymptote is y = 3/2 or y = 1.5.
Denominator coefficients: 1, -3, 1 (degree m = 2).
Since the degree of the numerator is less than the degree of the denominator (n < m), we are in Case 1.
Therefore, the horizontal asymptote is y = 0.
function parseCoefficients(coeffString) {
if (!coeffString) {
return [];
}
var coeffs = coeffString.split(',')
.map(function(c) { return parseFloat(c.trim()); })
.filter(function(c) { return !isNaN(c); });
return coeffs;
}
function getLeadingCoefficient(coeffs) {
for (var i = 0; i < coeffs.length; i++) {
if (coeffs[i] !== 0) {
return coeffs[i];
}
}
return 0; // Should ideally not happen for valid polynomials, but handle defensively
}
function getDegree(coeffs) {
for (var i = 0; i < coeffs.length; i++) {
if (coeffs[i] !== 0) {
return coeffs.length – 1 – i;
}
}
return -Infinity; // Represents the zero polynomial, degree is undefined or -Infinity
}
function calculateHorizontalAsymptote() {
var numeratorCoeffsString = document.getElementById("numeratorCoefficients").value;
var denominatorCoeffsString = document.getElementById("denominatorCoefficients").value;
var numeratorCoeffs = parseCoefficients(numeratorCoeffsString);
var denominatorCoeffs = parseCoefficients(denominatorCoeffsString);
var resultDiv = document.getElementById("result");
// Basic validation: Ensure we have at least one coefficient for each
if (numeratorCoeffs.length === 0 || denominatorCoeffs.length === 0) {
resultDiv.innerHTML = "Please enter coefficients for both numerator and denominator.";
return;
}
// Ensure the denominator polynomial is not the zero polynomial
var isDenominatorZero = denominatorCoeffs.every(function(c) { return c === 0; });
if (isDenominatorZero) {
resultDiv.innerHTML = "Denominator cannot be the zero polynomial.";
return;
}
var degreeN = getDegree(numeratorCoeffs);
var degreeM = getDegree(denominatorCoeffs);
var horizontalAsymptote = "";
if (degreeN degreeM
horizontalAsymptote = "No horizontal asymptote exists.";
}
resultDiv.innerHTML = horizontalAsymptote;
}