An asymptote is a straight line that a curve approaches as it heads toward infinity, describing the function’s behavior at its outer limits. While many rational functions approach horizontal or vertical lines, some follow a diagonal path instead. This diagonal path is known as a slant asymptote, and determining its equation is a straightforward process.
Understanding the Slant Asymptote
A slant asymptote, sometimes referred to as an oblique asymptote, is a straight line that is neither perfectly horizontal nor vertical. Graphically, the function’s curve gets progressively closer to this diagonal line as the $x$-values move far to the right (positive infinity) or far to the left (negative infinity). The function never actually touches or crosses the asymptote at these extreme ends, but the distance between the curve and the line approaches zero.
The existence of this specific type of asymptote depends entirely on the structure of the rational function. A rational function is defined as a fraction where both the numerator and the denominator are polynomials. A slant asymptote will only be present if the highest power of the variable in the numerator is exactly one unit greater than the highest power of the variable in the denominator.
If the degrees are equal, a horizontal asymptote exists; if the numerator’s degree is two or more greater, no linear asymptote exists. This degree comparison acts as the necessary pre-check before any calculation begins. This condition ensures that the result of the division will be a linear expression, which is the definition of a straight line.
The Calculation Method: Polynomial Division
Once the degree condition is met, the equation of the slant asymptote is found by performing polynomial division. This process involves dividing the numerator polynomial by the denominator polynomial, similar to how one performs long division with numbers. The result of this division will yield a quotient and a remainder, but only the quotient is relevant for finding the asymptote’s equation.
Polynomial long division is the most reliable method, as it works for any degree of denominator, unlike synthetic division which is limited to linear denominators. To begin the process, the terms of both the numerator and the denominator must be arranged in descending order of their exponents. Any missing powers of the variable should be included with a coefficient of zero to maintain proper alignment during the subtraction steps.
Consider the example function $\frac{x^2 + 2x + 3}{x – 1}$. The first step is to determine the term that, when multiplied by the divisor’s first term ($x$), results in the dividend’s first term ($x^2$). This term is $x$, which is placed above the division bar as the first term of the quotient.
Next, the term $x$ is multiplied by the entire divisor $(x – 1)$, yielding $x^2 – x$. This result is then subtracted from the dividend, which requires changing the signs of the terms being subtracted. Subtracting $(x^2 – x)$ from $(x^2 + 2x + 3)$ leaves a new polynomial of $3x + 3$.
The new polynomial, $3x + 3$, becomes the new dividend, and the process is repeated. The next term in the quotient is $+3$, since $3 \cdot x$ results in $3x$. Placing $+3$ in the quotient, we multiply it by the divisor $(x – 1)$ to get $3x – 3$.
Finally, $3x – 3$ is subtracted from $3x + 3$. Changing the signs and combining the terms results in a remainder of $6$. The division process stops when the degree of the remainder is less than the degree of the divisor, which is the case here since the remainder $6$ has a degree of zero and the divisor $x-1$ has a degree of one.
Extracting the Final Equation
The result of the polynomial division provides all the necessary information to write the final equation of the slant asymptote. The complete result of the division is expressed as the quotient plus the remainder divided by the divisor. However, for the purpose of finding the asymptote, the remainder term is entirely disregarded.
The slant asymptote is defined solely by the quotient polynomial obtained from the division. In the example of dividing $\frac{x^2 + 2x + 3}{x – 1}$, the quotient was $x + 3$. Therefore, the equation of the slant asymptote is $y = x + 3$.
This resulting equation will always be a linear function, taking the general form $y = mx + b$. The slope, $m$, and the $y$-intercept, $b$, define the diagonal line the function approaches. As $x$ approaches positive or negative infinity, the remainder term approaches zero. This means the function’s value becomes virtually indistinguishable from the quotient line, capturing the graph’s long-term behavior.