Horizontal and slant asymptotes describe the end behavior of a function, meaning what the graph does as x becomes very large positive or very large negative. They are especially important for rational functions because the highest degree terms control the graph far from the origin. Knowing the asymptote helps you sketch an accurate graph without plotting many points.
In calculus, this idea is expressed using limits at infinity.
Key Facts
- A horizontal asymptote y = L occurs if lim x->infinity f(x) = L or lim x->-infinity f(x) = L.
- For f(x) = P(x)/Q(x), if deg(P) < deg(Q), the horizontal asymptote is y = 0.
- For f(x) = P(x)/Q(x), if deg(P) = deg(Q), the horizontal asymptote is y = leading coefficient of P / leading coefficient of Q.
- For f(x) = P(x)/Q(x), if deg(P) = deg(Q) + 1, the slant asymptote is the quotient from polynomial long division.
- A slant asymptote has the form y = mx + b and occurs when lim x->infinity [f(x) - (mx + b)] = 0.
- If deg(P) is more than one greater than deg(Q), the function has a polynomial asymptote of degree greater than 1, not a slant asymptote.
Vocabulary
- Horizontal asymptote
- A horizontal line y = L that a graph approaches as x goes to positive or negative infinity.
- Slant asymptote
- A nonhorizontal line y = mx + b that a graph approaches as x goes to positive or negative infinity.
- End behavior
- The behavior of a function as x becomes very large positive or very large negative.
- Rational function
- A function that can be written as a ratio of two polynomials, f(x) = P(x)/Q(x), with Q(x) not equal to 0.
- Polynomial long division
- A method for rewriting a rational expression as a polynomial quotient plus a remainder over the original divisor.
Common Mistakes to Avoid
- Using vertical asymptote rules for horizontal asymptotes is wrong because vertical asymptotes come from zeros of the denominator, while horizontal asymptotes come from limits at infinity.
- Assuming every rational function has a horizontal asymptote is wrong because a slant asymptote occurs when the numerator degree is exactly one more than the denominator degree.
- Forgetting the leading coefficients when degrees are equal is wrong because y = a/b depends on the highest degree coefficients, not on the constant terms.
- Calling the quotient from long division the exact function is wrong because the rational function also includes a remainder term, which approaches 0 only as x goes to infinity or negative infinity.
Practice Questions
- 1 Find the horizontal asymptote of f(x) = (6x^3 - 2x + 5)/(3x^3 + 7x^2 - 1).
- 2 Find the slant asymptote of f(x) = (2x^2 + 5x - 3)/(x + 1) using polynomial long division.
- 3 Explain why f(x) = (x^3 + 2x)/(x + 4) does not have a horizontal or slant asymptote, and describe what kind of end behavior approximation it has instead.