AP Calculus AB/BC Unit 1 ReviewLimits and Continuity

AP Calculus Unit 1, Limits and Continuity, makes up 10-12% of the AP exam and builds the single idea everything else in calculus rests on. A function's value at a point and the value the function approaches near that point are two different things, and a limit measures the second one. You'll learn to find limits from graphs, tables, and algebra, classify discontinuities, and use limits to describe asymptotes and justify conclusions with the Intermediate Value Theorem.

What this unit covers

The limit idea and where it comes from

• The unit opens with a question that sounds philosophical but is actually mathematical. Can change happen at an instant? An average rate of change needs two points, and at a single instant the change in x is zero, so the formula breaks. Limits fix this by asking what the average rate of change approaches as the interval shrinks.
• The notation lim[x→c] f(x) = R means f(x) gets arbitrarily close to R when x gets sufficiently close to c, without x ever equaling c. The phrase "but not equal to c" is the whole point. The limit is about the neighborhood, not the point itself.
• One-sided limits look at the approach from one direction only. The left-hand limit lim[x→c⁻] f(x) and right-hand limit lim[x→c⁺] f(x) must match for the two-sided limit to exist.
• A limit can fail to exist in three classic ways. The function is unbounded near the point, the function oscillates wildly (like sin(1/x) near 0), or the one-sided limits disagree.

Finding limits four different ways

• From a graph, you trace the function toward the x-value from both sides and read off where the y-values head. Watch out for scale, since a graph can hide behavior that happens in a tiny window.
• From a table, you read the function values as the inputs close in on c from both directions. The values should settle toward one number.
• Algebraically, limit theorems let you split limits across sums, differences, products, quotients, and compositions. Direct substitution works whenever the function is continuous at the point.
• When substitution gives 0/0, you rewrite the expression first. Factor and cancel for rational functions, multiply by the conjugate for radicals, or swap in a trig identity. The original function and the rewritten one agree everywhere except possibly at the troublesome point, which is exactly where limits don't care.
• The Squeeze Theorem handles functions you can't simplify directly. Trap the function between two functions with the same limit and it gets forced to that limit too. This is how you establish lim[x→0] (sin x)/x = 1 and lim[x→0] (1 - cos x)/x = 0.

Continuity, defined precisely

• Informally, continuous means you can draw the graph without lifting your pencil. The formal three-part definition makes that checkable. A function f is continuous at x = c when f(c) exists, lim[x→c] f(x) exists, and the two are equal.
• Discontinuities come in three types. A removable discontinuity is a hole, where the limit exists but doesn't match the function value (or the function isn't defined). A jump discontinuity happens when the one-sided limits exist but disagree. A discontinuity due to a vertical asymptote happens when the function blows up.
• Polynomial, rational, power, exponential, logarithmic, and trigonometric functions are continuous at every point in their domains, so for those families "where is it discontinuous" really means "where is it not defined."
• If the limit exists at a hole, you can remove the discontinuity by defining or redefining the function's value to equal the limit. For piecewise functions, continuity at a boundary means the pieces must agree there, which often turns into solving for an unknown parameter like k.

Limits with infinity, in both directions

• Infinite limits describe what happens near a point where the function is unbounded. Writing lim[x→c] f(x) = ∞ says the outputs grow without bound, and a vertical asymptote at x = c is the graphical signature.
• Limits at infinity describe end behavior, what f(x) approaches as x → ∞ or x → -∞. If that limit is a finite number L, the graph has a horizontal asymptote y = L.
• For rational functions, compare degrees. Bigger degree on the bottom gives a horizontal asymptote at y = 0, equal degrees give the ratio of leading coefficients, bigger degree on top means no horizontal asymptote.
• Limits also let you compare growth rates. Exponentials eventually outgrow polynomials, and polynomials outgrow logarithms, and you can state that precisely with a limit of a ratio.

The Intermediate Value Theorem

• The IVT is the unit's first big existence theorem. If f is continuous on the closed interval [a, b] and d is any value between f(a) and f(b), then there is at least one c in (a, b) where f(c) = d.
• The classic use is guaranteeing a root. If f is continuous and f(a) is negative while f(b) is positive, the graph must cross zero somewhere in between.
• The justification format matters as much as the conclusion. You have to state that the function is continuous on the interval, show the endpoint values, and note that the target value sits between them. Skip the continuity statement and the argument earns nothing.

Unit 1, Limits and Continuity at a glance

Why Unit 1, Limits and Continuity matters in AP Calc

Limits are not a warm-up topic, they are the definition machine for the entire course. Every major object in calculus is literally defined as a limit, so the precision you build here pays off in every later unit.

• The derivative is the limit of average rates of change, so the question "can change occur at an instant?" from the first topic is answered in Unit 2 with the limit definition of the derivative.
• The definite integral is the limit of Riemann sums, so the "approach a value" idea returns when you accumulate change.
• The IVT is the first theorem where you justify a conclusion with hypotheses, which is the exact reasoning pattern the AP exam rewards on free-response questions all year.
• Continuity is the entry ticket for nearly every big theorem in the course, including the Mean Value Theorem and the Extreme Value Theorem, so knowing how to verify it is a skill you reuse constantly.

How this unit connects across the course

• The limit of the difference quotient becomes the formal definition of the derivative (Unit 2). When you compute lim[h→0] (f(x+h) - f(x))/h, you are doing Unit 1 algebra with Unit 2 meaning.
• Indeterminate forms come back with a new tool, L'Hospital's Rule, in contextual applications of differentiation (Unit 4). The factoring and conjugate techniques you learn now stay useful, but you get a shortcut once derivatives exist.
• The definite integral is defined as a limit of Riemann sums (Unit 6), and improper integrals in BC use limits at infinity directly (Unit 6).
• Infinite series convergence (Unit 10, BC only) is built entirely on limits of sequences of partial sums, and growth-rate comparisons from limits at infinity drive convergence tests like the ratio test.

Key formulas and procedures

• $\lim_{x \to c} f(x) = L$ means f(x) can be made arbitrarily close to L by taking x sufficiently close to c, but not equal to c. This is the working definition for the whole unit.
• Two-sided limit test: $\lim_{x \to c} f(x)$ exists if and only if $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x)$. Use it at every piecewise boundary and jump.
• Limit theorems: limits distribute over sums, differences, products, quotients (nonzero denominator limit), and compositions. These justify direct substitution for continuous functions.
• 0/0 toolkit: factor and cancel, multiply by the conjugate for radicals, or rewrite with trig identities, then substitute into the equivalent expression.
• Squeeze Theorem: if $g(x) \le f(x) \le h(x)$ near c and $\lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L$, then $\lim_{x \to c} f(x) = L$.
• Special trig limits: $\lim_{x \to 0} \frac{\sin x}{x} = 1$ and $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$. Memorize both and learn to manipulate expressions into these forms.
• Continuity definition: f is continuous at x = c when f(c) exists, $\lim_{x \to c} f(x)$ exists, and $\lim_{x \to c} f(x) = f(c)$. Cite all three parts when justifying.
• Removing a discontinuity: if the limit exists at a hole, redefine f(c) to equal that limit. For piecewise functions, set the one-sided limits equal at the boundary and solve for the parameter.
• Horizontal asymptotes of rational functions: compare degrees of numerator and denominator (smaller top gives y = 0, equal degrees give the ratio of leading coefficients, bigger top gives none).
• IVT: if f is continuous on [a, b] and d is between f(a) and f(b), then f(c) = d for at least one c in (a, b). Use it to guarantee that a function hits a specific value.

Unit 1, Limits and Continuity on the AP exam

Unit 1 carries 10-12% of the exam weight, and its skills show up far beyond questions explicitly labeled "limits."

• Multiple-choice questions test limits from every representation. You'll evaluate limits given a graph (often with holes, jumps, and asymptotes packed into one figure), estimate limits from a table of values, and compute limits algebraically where direct substitution gives 0/0.
• Piecewise continuity is a favorite format. A function is defined with an unknown constant, and you solve for the value that makes the pieces match at the boundary.
• Asymptote questions ask you to translate between limit statements and graph behavior in both directions. "Which limit statement describes the graph" and "find the horizontal asymptote" are standard.
• The IVT appears in free-response justification parts, often with a table of values rather than a formula. The scoring rewards the full argument, continuity stated, endpoint values shown, target value sandwiched between them. "The graph crosses zero" without citing continuity does not earn the point.
• Unit 1 reasoning is also embedded in later free-response work. Whenever you justify a claim using a theorem, you are using the hypotheses-then-conclusion habit this unit teaches.

Essential questions

• Can change occur at an instant, and how do limits make instantaneous rate of change meaningful?
• How can a function approach a value it never actually reaches, and why is that distinction useful?
• What does it mean, precisely, for a function to be continuous, and what kinds of breaks can occur?
• How does continuity let us guarantee that a function takes on a specific value without ever solving for it?

Key terms to know

• Limit: the value a function approaches as the input approaches some number, regardless of the function's actual value there.
• One-sided limit: the value a function approaches from only the left (x→c⁻) or only the right (x→c⁺).
• Indeterminate form: an expression like 0/0 that doesn't determine the limit by itself and signals you need to rewrite the function first.
• Squeeze Theorem: if a function is trapped between two functions that share a limit at a point, it must have that same limit.
• Continuity at a point: the three-part condition that f(c) exists, the limit at c exists, and the two values are equal.
• Removable discontinuity: a hole in the graph where the limit exists but the function value is missing or mismatched.
• Jump discontinuity: a break where the left and right one-sided limits both exist but are not equal.
• Vertical asymptote: a line x = c that the graph approaches as the function becomes unbounded, described by an infinite limit.
• Horizontal asymptote: a line y = L describing end behavior, given by a finite limit as x approaches ∞ or -∞.
• Infinite limit: a limit statement saying the function's outputs grow without bound near a point.
• Limit at infinity: a limit describing what the function approaches as the input grows without bound.
• Intermediate Value Theorem: the guarantee that a continuous function on [a, b] takes on every value between f(a) and f(b).
• Average rate of change: the change in output divided by the change in input over an interval, the slope of a secant line.
• Conjugate: the matching expression (like √x + 2 paired with √x - 2) you multiply by to clear radicals when evaluating a limit.

Common mix-ups

• The limit existing and the function being continuous are not the same thing. At a removable discontinuity, lim[x→c] f(x) exists just fine, but f isn't continuous there because f(c) doesn't match (or doesn't exist).
• L'Hospital's Rule is not a Unit 1 tool. It requires derivatives, which arrive in Unit 2 and get applied to indeterminate forms in Unit 4. On Unit 1 problems, use factoring, conjugates, identities, and the Squeeze Theorem.
• The IVT tells you a value c exists, not what c is, and not that there's only one. "At least one" is the exact guarantee.
• Writing lim[x→c] f(x) = ∞ describes unbounded behavior, but technically the limit does not exist as a real number. The infinity notation is a description, not a value.
• A vertical asymptote comes from an infinite limit at a point. A horizontal asymptote comes from a finite limit at infinity. Students swap these constantly; keep the "where x goes" straight.