To determine if a rational function has a slant (or oblique) asymptote, you need to look at the degrees of the polynomial in the numerator and the polynomial in the denominator.
A rational function is typically expressed in the form:
R(x) = P(x) / Q(x)
where:
- P(x) is the numerator (a polynomial)
- Q(x) is the denominator (also a polynomial)
For a rational function to have a slant asymptote, the degree of the numerator must be exactly one greater than the degree of the denominator. In other words:
- If degree(P) = degree(Q) + 1, then the function has a slant asymptote.
Here’s how you can find the slant asymptote:
- Perform polynomial long division or synthetic division of P(x) by Q(x).
- The quotient (the result of the division) will give you the equation of the slant asymptote.
For example, consider the rational function:
R(x) = (2x^3 + 3x^2 – 4) / (x^2 + 1)
The degree of the numerator, 2x^3 + 3x^2 – 4, is 3, and the degree of the denominator, x^2 + 1, is 2. Since the degree of the numerator is one greater than the degree of the denominator, we can conclude that this function has a slant asymptote.
In summary, check the degrees of the polynomials in the rational function. If the degree of the numerator is one greater than that of the denominator, then a slant asymptote exists, and you can find it through polynomial long division.