To find the slant (or oblique) asymptote of a rational function, follow these steps:
- Check the degrees of numerator and denominator: A slant asymptote occurs only if the degree of the numerator is exactly one greater than the degree of the denominator.
- Perform polynomial division: Divide the numerator by the denominator using long division (or synthetic division if applicable).
- Write the quotient as the asymptote: The quotient (ignoring the remainder) will be a linear polynomial of the form y=mx+by=mx+by=mx+b, which is the equation of the slant asymptote.
For example, if you have a function f(x)=2x2+3x+1x+2f(x)=\frac{2x^2+3x+1}{x+2}f(x)=x+22x2+3x+1, since the numerator degree (2) is one more than the denominator degree (1), perform long division:
2x2+3x+1÷(x+2)=2x−1 remainder 32x^2+3x+1\div (x+2)=2x-1\text{ remainder }32x2+3x+1÷(x+2)=2x−1 remainder 3
The slant asymptote is y=2x−1y=2x-1y=2x−1 (ignore the remainder)
Summary:
- Slant asymptote exists if deg(numerator)=deg(denominator)+1\deg(\text{numerator})=\deg(\text{denominator})+1deg(numerator)=deg(denominator)+1.
- Use long division of numerator by denominator.
- The quotient (linear polynomial) is the slant asymptote equation.
- Ignore the remainder after division.
This method shows how the function behaves like the line y=mx+by=mx+by=mx+b as xxx goes to infinity or negative infinity
