An infinite limit occurs when f(x) increases or decreases without bound as x approaches a specific value c, written lim x→c f(x) = ∞ or −∞. In AP Calculus (Topic 1.14), an infinite limit at x = c tells you the graph has a vertical asymptote there.
An infinite limit describes what happens when a function blows up. As x gets closer and closer to some value c, the outputs f(x) don't settle near a number. Instead they shoot toward positive or negative infinity. We write this as lim x→c f(x) = ∞ or lim x→c f(x) = −∞. Technically the limit "does not exist" as a real number, but the infinity notation is more useful because it tells you how it fails to exist. The function is unbounded, not oscillating or jumping.
This is the CED's extension of the limit concept (EK LIM-2.D.1), and it's the precise language calculus uses to describe asymptotic and unbounded behavior (EK LIM-2.D.2). The classic example is f(x) = 1/x near x = 0. From the right, the values climb toward ∞; from the left, they plunge toward −∞. One-sided limits matter a lot here, because a function can head to +∞ on one side of c and −∞ on the other. Whenever you get an infinite limit at x = c, the graph has a vertical asymptote at x = c.
Infinite limits live in Unit 1 (Limits and Continuity), specifically Topic 1.14, Connecting Infinite Limits and Vertical Asymptotes. The learning objective is 1.14.A, which asks you to interpret the behavior of functions using limits involving infinity. This is the moment in AP Calc where limit notation stops being abstract and starts describing actual graph features. An infinite limit IS a vertical asymptote, just written in limit language instead of drawn as a dashed line. Beyond Unit 1, this idea keeps paying off. You'll use it to classify discontinuities, to justify why a function is unbounded on an interval, and later to set up improper integrals in Unit 6, where integrating across a vertical asymptote forces you back to limit notation.
Keep studying AP Calculus Unit 1
Visual cheatsheet
view galleryVertical Asymptote (Unit 1)
These two are the same fact in two languages. The graph picture is the vertical asymptote, and the algebra sentence is the infinite limit. If lim x→c⁺ f(x) or lim x→c⁻ f(x) is ±∞, then x = c is a vertical asymptote, and the exam expects you to translate fluently in both directions.
Horizontal Asymptote (Unit 1)
Horizontal asymptotes come from the mirror-image concept, limits AT infinity. There, x runs off to ±∞ and f(x) settles toward a finite value L. Keeping these two straight (infinite output vs. infinite input) is one of the most common Unit 1 stumbling points.
Removable Discontinuity vs. Unbounded Behavior (Unit 1)
Not every zero in a denominator produces an infinite limit. In g(x) = (x³ − 8)/(x − 2), the factor (x − 2) cancels, so the limit at x = 2 is finite (it equals 12) and the graph has a hole, not an asymptote. Always check whether the troublesome factor cancels before declaring an infinite limit.
Exponential Growth/Decay (Units 6-7)
Exponential functions also grow without bound, but as x → ∞, not at a single point. Comparing the two kinds of unbounded behavior sharpens your sense of when to write lim x→c f(x) = ∞ versus lim x→∞ f(x) = ∞.
Infinite limits show up mostly as multiple-choice questions in two flavors. The first gives you a function (often rational or trig-over-polynomial, like sin x / (x² − π²)) and asks you to identify where it has vertical asymptotes or to evaluate a one-sided limit there, including the sign of the infinity. The second goes the other way. You're told a function has a vertical asymptote at some x-value and asked which limit statement describes the behavior, so you need the vocabulary, not just the computation. The trap to watch for is the removable discontinuity. A question like the one on g(x) = (x³ − 8)/(x − 2) is designed to catch people who see a zero denominator and instantly write ∞, when the factor actually cancels and the limit is finite. No released FRQ centers on this term by name, but FRQs routinely expect correct limit notation when you justify unbounded behavior or evaluate improper integrals later in the course.
An infinite limit and a limit at infinity sound identical but are opposites. In an infinite limit, x approaches a finite value c and the OUTPUT blows up (lim x→c f(x) = ∞), giving a vertical asymptote. In a limit at infinity, the INPUT blows up (x → ∞) and the output often settles toward a finite value L, giving a horizontal asymptote. Quick check: where is the infinity symbol? Under the lim means limit at infinity; on the answer side means infinite limit.
An infinite limit means f(x) grows or decreases without bound as x approaches a finite value c, written lim x→c f(x) = ∞ or −∞.
An infinite limit at x = c is exactly equivalent to a vertical asymptote at x = c, which is the whole point of Topic 1.14.
Always check one-sided limits separately, because a function can go to +∞ from one side of c and −∞ from the other.
A zero in the denominator does not guarantee an infinite limit; if the factor cancels (like x − 2 in (x³ − 8)/(x − 2)), you get a hole and a finite limit instead.
Saying the limit 'equals infinity' is a precise way of saying the limit does not exist because the function is unbounded, which is different from oscillating or jump behavior.
Don't confuse infinite limits (x → c, output unbounded) with limits at infinity (x → ∞, which describe end behavior and horizontal asymptotes).
It's when f(x) increases or decreases without bound as x approaches a specific value c, written lim x→c f(x) = ∞ or −∞. It's covered in Topic 1.14 of Unit 1 and signals a vertical asymptote at x = c.
Strictly speaking, no. The limit does not exist as a real number because the function is unbounded. But writing lim x→c f(x) = ∞ is correct notation on the AP exam because it describes exactly how the limit fails to exist.
In an infinite limit, x approaches a finite number and f(x) blows up, giving a vertical asymptote. In a limit at infinity, x itself goes to ±∞ and you're describing end behavior, which is how horizontal asymptotes are defined. Check where the ∞ symbol sits to tell them apart.
No. If the factor that makes the denominator zero also cancels from the numerator, you get a removable discontinuity (a hole) and a finite limit. For example, (x³ − 8)/(x − 2) has a finite limit of 12 at x = 2, not a vertical asymptote.
They're two descriptions of the same thing. If any one-sided limit of f(x) at x = c equals ∞ or −∞, then x = c is a vertical asymptote. That translation between limit notation and graph behavior is exactly what learning objective 1.14.A tests.