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Unit 1 · Limits & Continuity Flashcards Cheat Sheet Essentials Visual Review MC Practice FRQ Practice

AP Calculus AB/BC Unit 1 Essentials

The must-know terms and core concepts for Unit 1: Limits & Continuity. Every vocabulary word and idea you need to master.

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Key Concept 1
A limit describes what a function approaches — the foundation of calculus
A limit, lim(x→c) f(x) = L, captures the value f(x) approaches as x gets close to c, regardless of what happens exactly at c. One-sided limits describe approach from the left or right, and the two-sided limit exists only when the one-sided limits agree. This idea — approaching a value arbitrarily closely — is what makes it possible to talk about instantaneous rates of change and the area under a curve later in the course.
Limits One-Sided Limits Existence
Key Concept 2
Evaluate limits by substitution, then algebra when needed
If a function is continuous at c, you find the limit by direct substitution. When substitution gives the indeterminate form 0/0, you must simplify first — usually by factoring and canceling or rationalizing with the conjugate — and then substitute. The Squeeze Theorem handles limits you can bound between two functions with the same limit, and the special limit lim(x→0) sin x / x = 1 appears throughout calculus.
Substitution Algebraic Techniques Squeeze Theorem
Key Concept 3
Continuity connects limits to graphs, asymptotes, and the IVT
A function is continuous at c when f(c) is defined, the limit exists, and the two are equal; failing any part gives a removable, jump, or infinite discontinuity. Limits also explain asymptotes: infinite limits produce vertical asymptotes, and limits at infinity produce horizontal asymptotes. Finally, the Intermediate Value Theorem uses continuity on a closed interval to guarantee that a function attains every value between its endpoints' outputs.
Continuity Discontinuities & Asymptotes IVT
Limit
The value f(x) approaches as x approaches c, written lim(x→c) f(x) = L.
Limits
One-sided limit
The value f(x) approaches from the left (c⁻) or the right (c⁺) only.
Limits
Existence of a limit
A two-sided limit exists if and only if the left and right limits exist and are equal.
Limits
Limit laws
Rules that let limits distribute over sums, products, quotients, and constant multiples.
Evaluating
Direct substitution
Evaluating a limit by plugging in x = c when f is continuous there.
Evaluating
Indeterminate form (0/0)
A result of substitution that requires algebraic simplification before the limit can be found.
Evaluating
Factoring
Canceling a common factor that causes 0/0, then substituting.
Evaluating
Rationalizing
Multiplying by a conjugate to remove a radical causing 0/0.
Evaluating
Squeeze Theorem
If g ≤ f ≤ h near c and lim g = lim h = L, then lim f = L.
Evaluating
sin x / x limit
lim(x→0) sin x / x = 1, a fundamental trigonometric limit.
Evaluating
Continuity at a point
f(c) is defined, the limit exists, and the limit equals f(c).
Continuity
Continuity over an interval
Continuity holding at every point of an interval.
Continuity
Removable discontinuity
A hole where the limit exists but f(c) is undefined or unequal to it.
Discontinuities
Jump discontinuity
A point where the one-sided limits exist but differ.
Discontinuities
Infinite discontinuity
A point where the function grows without bound (a vertical asymptote).
Discontinuities
Infinite limit
lim(x→c) f(x) = ±∞, indicating unbounded behavior and a vertical asymptote.
Asymptotes
Vertical asymptote
A line x = c that the graph approaches as the limit tends to ±∞.
Asymptotes
Limit at infinity
lim(x→±∞) f(x) = L, describing the function's end behavior.
Asymptotes
Horizontal asymptote
A line y = L given by a finite limit at infinity.
Asymptotes
Intermediate Value Theorem
If f is continuous on [a, b] and k lies between f(a) and f(b), then f(c) = k for some c in [a, b].
IVT
Average rate of change
The slope of the secant line over an interval, [f(b) − f(a)]/(b − a).
Foundations