AP® Calculus AB · Unit 1 · Topic 1.15
Connecting Limits at Infinity and Horizontal Asymptotes
A horizontal asymptote is the height a function's limit settles toward as x goes to infinity or negative infinity, found by dividing every term by the highest power of x in the denominator.
As x grows huge, the terms in a function that grow fastest completely dominate the terms that grow slower. In a ratio of polynomials, the highest-power terms on top and bottom are what decide the end behavior, because every smaller term becomes insignificant by comparison. That dominant behavior is what produces a flat height the graph settles toward, called a horizontal asymptote.
To see it cleanly, divide every single term, top and bottom, by the highest power of x that appears in the denominator. Each lower-order term turns into something like 5/x2 or 1/x, and as x heads to infinity those fractions shrink to zero. What survives is a clean ratio of leading coefficients, and that ratio is the limit. The same trick works heading toward negative infinity, since those vanishing terms die off regardless of which direction x runs.
This connects directly to end behavior of rational functions generally. When the degree on top matches the degree on bottom, you get a finite horizontal asymptote, the ratio of leading coefficients. When the top degree is larger, the fraction grows without bound instead of leveling off, so there is no horizontal asymptote — that is where slant or curved end behavior takes over, a preview of behavior you'll analyze more closely later.
Key terms
- Horizontal asymptote
- A horizontal line y = L that a function's graph approaches as x goes to infinity or negative infinity.
- Limit at infinity
- The value a function approaches as x increases or decreases without bound.
- Leading term
- The term in a polynomial with the highest power of x, which dominates end behavior.
- End behavior
- How a function's output behaves as the input grows very large in the positive or negative direction.
Watch it explained
- How it worksHow does dividing every term by the highest power reveal the end behavior?
- Worked exampleFind the limit of (3x2 + 5)/(2x2 - x) as x heads to infinity.
- Common mistakeWhy isn't a horizontal asymptote a line the curve is forbidden to cross?
- Going furtherWhat happens to the horizontal asymptote when the top degree beats the bottom degree?
Check yourself
End of topic.