How do you find slant asymptotes using limits?

To find slant asymptotes for a rational function, we need to analyze the behavior of the function as the variable approaches infinity. Here’s how to do it:

1. Start with the function: Consider a rational function of the form f(x) = p(x)/q(x), where the degree of the numerator p(x) is exactly one more than the degree of the denominator q(x).
2. Perform polynomial long division: Divide p(x) by q(x). The result will be in the form f(x) = mx + b + r(x), where mx + b is the quotient representing the slant asymptote and r(x) is the remainder.
3. Identify the dominant term: As x approaches infinity, the remainder r(x) will become negligible. Therefore, the behavior of f(x) will be similar to that of the line y = mx + b.
4. Write the slant asymptote: The slant asymptote can be expressed as y = mx + b, where m is the leading coefficient from the division, and b is the constant term from the quotient.

In summary, by using polynomial long division, you can effectively find the slant asymptote of a rational function as you analyze its limits at infinity. This process helps to understand the end behavior of the function.