Unit 1 · Topic 1.15

1.15 Connecting Limits at Infinity and Horizontal Asymptotes

A limit at infinity describes what f(x) does as x grows without bound in the positive or negative direction, called end behavior. If f(x) approaches a number L, the line y = L is a horizontal asymptote. For rational functions, comparing the highest powers gives the answer fast.

Key terms

limit at infinity
end behavior
horizontal asymptote
dominant term
degree

Limits at infinity and horizontal asymptotes

lim (x→∞) f(x) = L means f(x) gets as close as you like to L once x is large enough. The line y = L is then a horizontal asymptote. Check x→−∞ separately; a function can have a different horizontal asymptote on each end, or one on just one end.

Unlike vertical asymptotes, a graph is allowed to cross a horizontal asymptote. For example, cos x / x crosses y = 0 infinitely often while still approaching it.

Rational functions: compare the degrees

Divide every term by the highest power of x in the denominator. Terms like 5/x or 3/x² go to 0 as x→±∞, which leaves the answer. The shortcut:

Degree comparison	Limit as x→±∞	Example
Top degree < bottom degree	0	(3x + 1)/(x² + 4) → 0
Top degree = bottom degree	Ratio of leading coefficients	(3x² − 5x)/(7 − 2x²) → −3/2
Top degree > bottom degree	∞ or −∞ (no horizontal asymptote)	x³/(x² + 1) → ∞ as x→∞

Dominant terms beyond polynomials

As x→∞, some functions grow much faster than others. From slowest to fastest: ln x, then powers like √x or x², then exponentials like eˣ. The fastest-growing term dominates a sum or quotient.

Facts worth knowing: lim (x→∞) eˣ = ∞ and lim (x→−∞) eˣ = 0, so y = 0 is a horizontal asymptote of eˣ on the left. lim (x→∞) e⁻ˣ = 0. lim (x→∞) ln x = ∞, slowly but without bound. Also lim (x→±∞) 1/xⁿ = 0 for any n > 0.

Example: lim (x→∞) (2eˣ + 1) / (eˣ − 3). Divide top and bottom by eˣ: (2 + e⁻ˣ) / (1 − 3e⁻ˣ) → 2/1 = 2. But as x→−∞, eˣ→0, so the expression approaches (0 + 1) / (0 − 3) = −1/3. That function has two different horizontal asymptotes, y = 2 and y = −1/3.

Square roots and negative infinity

√(x²) = |x|, not x. When x→−∞, |x| = −x. So when you pull x² out of a square root for negative x, a minus sign appears. This is why some functions have two different horizontal asymptotes.

Worked examples

Try each one yourself first, then open the solution.

Example 1
Equal degrees
Find lim (x→∞) (3x² − 5x) / (7 − 2x²).
Show the solutionHide the solution
1. Step 1: The top and bottom both have degree 2. Divide every term by x²: (3 − 5/x) / (7/x² − 2).
2. Step 2: As x→∞, 5/x → 0 and 7/x² → 0.
3. Step 3: The limit is 3 / (−2).
Answer: −3/2, so y = −3/2 is a horizontal asymptote.
Example 2
Trap: a square root as x→−∞
Find lim (x→∞) 4x / √(x² + 1) and lim (x→−∞) 4x / √(x² + 1).
Show the solutionHide the solution
1. Step 1: Factor x² inside the root: √(x² + 1) = √(x²)·√(1 + 1/x²) = |x|·√(1 + 1/x²).
2. Step 2: As x→∞, |x| = x, so the expression is 4x / (x√(1 + 1/x²)) = 4 / √(1 + 1/x²) → 4/1 = 4.
3. Step 3: As x→−∞, |x| = −x, so the expression is 4x / (−x√(1 + 1/x²)) = −4 / √(1 + 1/x²) → −4.
4. Step 4: Sense check: for large negative x, the top is negative and the bottom (a square root) is positive, so the answer must be negative.
Answer: lim (x→∞) = 4 and lim (x→−∞) = −4. The graph has two horizontal asymptotes, y = 4 and y = −4.

Common mistakes

Writing √(x²) = x for negative x. It equals |x|, which is −x when x < 0.
Assuming a graph can never cross a horizontal asymptote. It can; the asymptote only describes end behavior.
Checking only x→∞. The limit as x→−∞ can be different.

On the exam

Multiple-choice questions ask for horizontal asymptotes of rational functions, often with the terms written out of order, so find the highest-degree terms carefully.
In context problems, lim (t→∞) describes the long-run value of a model, like a population leveling off. Interpret it with units.

Connected topics

Videos

Calculus AB/BC – 1.15 Limits at Infinity and Horizontal Asymptotes
The AlgebrosWatch on YouTube (opens in a new tab)
Limits at infinity of quotients (Part 1) | Limits and continuity | AP Calculus AB | Khan Academy
Khan AcademyWatch on YouTube (opens in a new tab)
AP Calculus AB TOPIC 1.15 Connecting Limits at Infinity and Horizontal Asymptotes
Math Teacher GOATWatch on YouTube (opens in a new tab)
Limits at Infinity & Horizontal Asymptotes
The Organic Chemistry TutorWatch on YouTube (opens in a new tab)
Computing Limits at Infinity for Rational Functions
Dr. Trefor BazettWatch on YouTube (opens in a new tab)
Limits at infinity of quotients (Part 2) | Limits and continuity | AP Calculus AB | Khan Academy
Khan AcademyWatch on YouTube (opens in a new tab)