A limit at infinity is the value a function approaches as x increases (or decreases) without bound, written lim as x→∞ of f(x). On the AP Calculus exam it describes a function's end behavior, and when the limit equals a finite number L, the graph has a horizontal asymptote at y = L.
A limit at infinity answers a simple question. If you let x get bigger and bigger forever, where do the function's outputs head? If f(x) settles toward a single number L, we write lim as x→∞ of f(x) = L. If the outputs keep growing without bound, the limit is ∞ (or −∞ if they plummet). The CED frames this as an extension of the basic limit concept (EK LIM-2.D.3), and its whole job is to describe end behavior, meaning what the graph does way out on the left and right edges (EK LIM-2.D.4).
The workhorse skill here is comparing growth rates (EK LIM-2.D.5). For a rational function, you're really asking which wins the race as x explodes, the numerator or the denominator. If the denominator grows faster, the limit is 0. If they grow at the same rate, the limit is the ratio of leading coefficients. If the numerator grows faster, the limit is infinite. That one comparison instinct answers most limit-at-infinity questions you'll see.
This term lives in Topic 1.15 of Unit 1 (Limits and Continuity) and directly supports learning objective 1.15.A, interpreting the behavior of functions using limits involving infinity. It's the bridge between limit notation and something you can see on a graph. A finite limit at infinity is a horizontal asymptote, so this concept turns an algebraic computation into a graphical fact. It also plants the growth-rate comparison idea (exponentials beat polynomials, polynomials beat logs) that resurfaces all over the course, from L'Hospital-style indeterminate forms to improper integrals and series convergence in BC.
Keep studying AP® Calculus Unit 1
Visual cheatsheet
view galleryEnd Behavior (Unit 1)
Limits at infinity are the formal language for end behavior. When a precalc teacher said "the graph levels off at y = 2," calculus says lim as x→∞ of f(x) = 2. Same idea, now with notation you can compute with.
Horizontal Asymptotes (Unit 1, Topic 1.15)
A horizontal asymptote is just a finite limit at infinity drawn on the graph. If the limit as x→∞ or x→−∞ equals L, the line y = L is a horizontal asymptote. Check both directions, because a function can have two different ones.
L'Hospital's Rule (Unit 4)
When a limit at infinity gives you ∞/∞, L'Hospital's Rule lets you compare rates of change directly by differentiating top and bottom. It's EK LIM-2.D.5's growth-rate idea upgraded with derivatives.
Improper Integrals (Unit 6, BC)
BC evaluates integrals with infinite bounds by taking a limit at infinity of the antiderivative. Whether the area under a curve out to infinity is finite comes down to exactly the end-behavior reasoning you learn here.
This shows up constantly in multiple choice, usually in two flavors. Flavor one hands you a rational function like (2x² + 10x)/(3x² + 5) and asks for the limit as x→∞ (answer: 2/3, since the degrees match, so take the ratio of leading coefficients). Flavor two asks it backwards, like "which function has a horizontal asymptote at y = 3?", and you have to recognize that's secretly a limit-at-infinity question. You also need to spot when the answer is infinite, like lim as x→∞ of x³ + 10x − 42, where the function just grows without bound. No released FRQ centers on this term by name, but free-response answers that justify a horizontal asymptote must cite a limit at infinity to earn the point. "The degrees are equal" is not a justification; "lim as x→∞ f(x) = 2/3" is.
These sound identical but point opposite directions. A limit AT infinity asks what y-value f approaches as x runs off to ±∞, and it describes end behavior and horizontal asymptotes. An INFINITE limit happens at a finite x-value where f(x) blows up to ±∞, and it describes vertical asymptotes. Quick check: is infinity under the lim (input) or on the right side of the equals sign (output)? Input means limit at infinity, output means infinite limit.
A limit at infinity tells you what value f(x) approaches as x grows without bound, and it formally describes a function's end behavior (EK LIM-2.D.4).
If lim as x→∞ of f(x) = L for a finite number L, the function has a horizontal asymptote at y = L, so finding horizontal asymptotes means computing limits at infinity.
For rational functions, compare degrees. Bottom-heavy means the limit is 0, equal degrees means the ratio of leading coefficients, and top-heavy means the limit is ±∞.
Always check x→∞ and x→−∞ separately, since a function can approach different values (or no value) in each direction.
Limits at infinity let you compare relative magnitudes and rates of growth of functions (EK LIM-2.D.5), an idea that returns in L'Hospital's Rule and BC's improper integrals.
It's the value a function approaches as x increases or decreases without bound, written lim as x→∞ of f(x). It's covered in Topic 1.15 of Unit 1 and describes a function's end behavior.
No. A limit at infinity has infinity as the input (x→∞) and relates to horizontal asymptotes, while an infinite limit has infinity as the output (f(x)→∞ at a finite x-value) and relates to vertical asymptotes.
No. The limit can be a finite number L (giving a horizontal asymptote at y = L), it can be ∞ or −∞ (like x³ + 10x − 42 as x→∞), or it can fail to exist entirely, like sin(x), which oscillates forever without settling.
Compare the degrees of the numerator and denominator. If the denominator's degree is higher the limit is 0, if the degrees are equal it's the ratio of leading coefficients (so (2x² + 10x)/(3x² + 5) goes to 2/3), and if the numerator's degree is higher the limit is ±∞.
They're two views of the same fact. The line y = L is a horizontal asymptote exactly when the limit of f(x) as x→∞ or x→−∞ equals L, which is why AP justifications for horizontal asymptotes must be written as limits.
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