To find the slant (or oblique) asymptote of a rational function, you need to perform polynomial long division when the degree of the numerator is exactly one greater than the degree of the denominator.
Here’s a step-by-step guide:
- Identify the Rational Function: Start with a rational function in the form f(x) = P(x) / Q(x), where P(x) is the numerator and Q(x) is the denominator.
- Check the Degrees: Ensure that the degree of P(x) (numerator) is one more than the degree of Q(x) (denominator). If this condition is not met, there is no slant asymptote.
- Perform Polynomial Long Division: Divide P(x) by Q(x). The quotient you get (disregarding the remainder) will be the equation of the slant asymptote.
- Write the Equation: The result from the long division will give you a linear equation in the form of y = mx + b, which is the equation of the slant asymptote.
Example: Consider the function f(x) = (2x^2 + 3x + 1) / (x + 1). The degree of the numerator (2) is one greater than the degree of the denominator (1). Dividing 2x^2 + 3x + 1 by x + 1 gives us 2x + 1 as the quotient, which is the slant asymptote.
In summary, by ensuring the correct degree relationship and performing long division, you can successfully find the slant asymptotes of rational functions.