Review AP Calculus AB/BC Unit 1 to build the limit and continuity foundation every later unit depends on. From estimating limits graphically and numerically to applying the Intermediate Value Theorem, this unit introduces the core reasoning tools of calculus.
Use the topic guides, key terms, and practice questions available on this page to work through all 16 topics before your exam.
Calculus begins with a deceptively simple question: what value does a function approach as its input gets close to some number? Unit 1 answers that question rigorously using limits, then uses limits to define continuity and to prove the Intermediate Value Theorem.
Unit 1 is about limits and continuity. You learn to express, estimate, and evaluate limits using graphs, tables, algebra, and the Squeeze Theorem, then use the formal definition of continuity to classify discontinuities, confirm continuity over intervals, remove removable discontinuities, and apply the IVT.
A limit lim[x→c] f(x) = R describes what output a function approaches as x gets close to c, independent of the actual value f(c). You can estimate limits from graphs and tables, or evaluate them exactly using limit laws, algebraic manipulation such as factoring and conjugate rationalization, and the Squeeze Theorem for cases like lim[x→0] sin(x)/x = 1.
A function is continuous at x = c when f(c) exists, lim[x→c] f(x) exists, and those two values are equal. Discontinuities are classified as removable (hole), jump (unequal one-sided limits), or infinite (vertical asymptote). Polynomials, rationals, exponentials, logarithms, and trig functions are continuous on their entire domains.
Infinite limits describe unbounded behavior near a vertical asymptote: lim[x→a] f(x) = ±∞. Limits at infinity describe end behavior: if lim[x→∞] f(x) = L, then y = L is a horizontal asymptote. For rational functions, compare degrees of numerator and denominator to determine the horizontal asymptote or confirm none exists.
The derivative in Unit 2 is defined as lim[h→0] (f(a+h) - f(a))/h, a limit of an average rate of change. The definite integral in Unit 6 is a limit of Riemann sums. For BC students, infinite series in Unit 10 rely on limits of partial sums. Every major concept in calculus is built on the limit idea introduced here.
Introduces the central question of calculus: how to define the rate of change at a single point using limits of average rates of change over shrinking intervals.
Defines the limit formally as the value f(x) approaches as x gets close to c, and introduces the notation lim[x→c] f(x) = R across graphical, numerical, and analytic representations.
Covers reading one-sided and two-sided limits from graphs, identifying DNE cases due to unbounded behavior, oscillation, or unequal one-sided limits, and recognizing scale issues.
Uses tables of x-values approaching c from both sides to estimate limits numerically, including recognizing convergence, divergence, and DNE from tabular data.
Applies limit laws for sums, differences, products, quotients, and composite functions to evaluate limits analytically, including one-sided limits.
Resolves 0/0 indeterminate forms by factoring and canceling, multiplying by a conjugate, or rewriting trig expressions before evaluating the limit.
Focuses on choosing the correct technique (direct substitution, factoring, conjugate, dominant-term analysis) based on the structure of the limit expression.
Uses the Squeeze Theorem to evaluate limits of bounded or oscillatory functions, with key results lim[x→0] sin(x)/x = 1 and lim[x→0] (1 - cos x)/x = 0.
Practices translating limit information across graphical, numerical, and analytic forms, reinforcing that all three representations describe the same limit behavior.
Classifies discontinuities as removable (hole), jump (unequal one-sided limits), or infinite (vertical asymptote) using one-sided limits and function values.
States the three-condition definition of continuity at x = c: f(c) exists, the limit exists, and they are equal. Identifies which condition fails for each discontinuity type.
Determines intervals of continuity by checking domain restrictions and boundary points, using the fact that standard function families are continuous on their domains.
Removes removable discontinuities by redefining f(c) to equal the limit, and finds parameter values that make piecewise functions continuous at boundary points.
Links infinite limits to vertical asymptotes, using sign analysis on both sides of x = a to determine whether the function approaches +∞ or -∞.
Uses limits at infinity to describe end behavior and identify horizontal asymptotes, applying degree comparison for rational functions and growth-rate comparisons.
Applies the IVT to guarantee the existence of a function value on a closed interval, requiring explicit verification of continuity and the target value being between f(a) and f(b).
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Review Determining Limits Using the Squeeze Theorem with attention to how the concept appears in AP-style source and evidence questions.
Review Determining Limits Using Algebraic Properties of Limits with attention to how the concept appears in AP-style source and evidence questions.
Review Defining Continuity at a Point with attention to how the concept appears in AP-style source and evidence questions.
Review Confirming Continuity over an Interval with attention to how the concept appears in AP-style source and evidence questions.
Calculus asks whether change can happen at a single instant. The average rate of change over an interval [a, a+h] is Δy/Δx = (f(a+h) - f(a))/h, which is the slope of a secant line. As h shrinks toward zero, the secant line approaches the tangent line, and the limit of that ratio defines the instantaneous rate of change at x = a. This is the conceptual origin of the derivative.
Given f(x) = x², write the average rate of change from x = 2 to x = 2+h and describe what happens as h → 0.
The limit lim[x→c] f(x) = R means f(x) can be made arbitrarily close to R by taking x sufficiently close to c, without requiring x to equal c. Limits can be expressed graphically (reading where the graph heads), numerically (building a table of values approaching c from both sides), or analytically (using notation and algebra). A two-sided limit exists only when the left-hand limit and right-hand limit both exist and are equal. A limit fails to exist when the function is unbounded near c, oscillates (like sin(1/x) near 0), or has unequal one-sided limits.
A graph shows an open circle at (3, 5) and a filled dot at (3, 2). What is lim[x→3] f(x)? What is f(3)?
| Representation | How to estimate the limit | Key caution |
|---|---|---|
| Graph | Read the y-value both sides approach as x → c | Open vs. filled circles show f(c), not the limit |
| Table | Check that outputs from left and right converge to the same value | Outputs may converge slowly; use values very close to c |
| Analytic | Use limit notation and algebraic or theorem-based evaluation | Direct substitution may give 0/0; further work needed |
Limit laws let you break complex limits into simpler pieces: the limit of a sum, difference, product, or quotient equals the corresponding operation on the individual limits (provided the limits exist and denominators are nonzero). Direct substitution works whenever the function is continuous at c. When substitution gives the indeterminate form 0/0, use algebraic manipulation: factor and cancel common factors in rational functions, multiply by a conjugate to simplify radical expressions, or use alternate trigonometric forms. Topic 1.7 focuses on choosing the right technique for a given limit.
Evaluate lim[x→3] (x² - 9)/(x - 3) by factoring. Then identify which limit law justifies evaluating lim[x→2] [f(x) + g(x)] when both individual limits exist.
The Squeeze Theorem states that if g(x) ≤ f(x) ≤ h(x) near c and lim[x→c] g(x) = lim[x→c] h(x) = L, then lim[x→c] f(x) = L. The two most important results are lim[x→0] sin(x)/x = 1 and lim[x→0] (1 - cos x)/x = 0, both proved by squeezing. Topic 1.9 asks you to move fluidly between graphical, numerical, and analytic representations of the same limit, recognizing that all three describe the same underlying behavior.
Use the Squeeze Theorem to find lim[x→0] x·sin(1/x), given that -|x| ≤ x·sin(1/x) ≤ |x|.
A function is continuous at x = c when all three conditions hold: f(c) exists, lim[x→c] f(x) exists, and lim[x→c] f(x) = f(c). Failing any one condition produces a discontinuity. Removable discontinuities (holes) occur when the limit exists but either f(c) is undefined or f(c) does not equal the limit. Jump discontinuities occur when the one-sided limits exist but are unequal. Infinite discontinuities occur when the limit is ±∞, corresponding to a vertical asymptote.
For f(x) = (x² - 4)/(x - 2), identify the type of discontinuity at x = 2 and state which of the three continuity conditions fails.
| Type | One-sided limits | f(c) | Removable? |
|---|---|---|---|
| Removable (hole) | Both exist and are equal | Undefined or wrong value | Yes, redefine f(c) |
| Jump | Both exist but are unequal | May or may not exist | No |
| Infinite (vertical asymptote) | One or both are ±∞ | Undefined | No |
A function is continuous on an interval when it is continuous at every point in that interval, with appropriate one-sided conditions at closed endpoints. Polynomials are continuous everywhere. Rational functions are continuous wherever the denominator is nonzero. Exponential, logarithmic, and trigonometric functions are continuous on their domains. To remove a removable discontinuity, redefine f(c) to equal the limit at that point. For piecewise functions, continuity at a boundary requires the expressions on both sides to produce the same value at the boundary point, which often means solving for an unknown parameter.
Find the value of k that makes f(x) = {kx + 1, x < 2; x² - 1, x ≥ 2} continuous at x = 2.
Infinite limits describe behavior near a vertical asymptote: if lim[x→a⁺] f(x) = +∞, the graph rises without bound just to the right of x = a. Check both one-sided limits because the signs can differ. Limits at infinity describe end behavior: lim[x→∞] f(x) = L means the graph levels off toward y = L, a horizontal asymptote. For rational functions, compare the degrees of numerator and denominator: equal degrees give a horizontal asymptote at the ratio of leading coefficients; lower numerator degree gives y = 0; higher numerator degree means no horizontal asymptote. Relative magnitudes of growth rates matter: exponential functions grow faster than power functions, which grow faster than logarithms.
Find all vertical and horizontal asymptotes of f(x) = (3x² + 1)/(x² - 4) using limits.
| Asymptote type | Limit form | What it describes |
|---|---|---|
| Vertical | lim[x→a] f(x) = ±∞ | Unbounded behavior near x = a |
| Horizontal | lim[x→∞] f(x) = L | End behavior as x grows large |
| None (rational) | Numerator degree > denominator degree | No horizontal asymptote; may have slant asymptote |
The IVT states: if f is continuous on the closed interval [a, b] and d is any number strictly between f(a) and f(b), then there exists at least one c in (a, b) such that f(c) = d. The most common application is proving a root exists: if f(a) and f(b) have opposite signs and f is continuous on [a, b], then f(c) = 0 for some c in (a, b). The IVT guarantees existence but does not find c or say c is unique. Always state continuity on the closed interval before invoking the theorem.
Show that f(x) = x³ - x - 1 has a root on [1, 2] by applying the IVT. State each condition explicitly.
Try AP-style multiple-choice questions and written prompts after you review the notes.
| Term | Definition |
|---|---|
| limit at infinity | The value a function approaches as x increases or decreases without bound; written lim[x→∞] f(x) = L or lim[x→-∞] f(x) = L. |
| Two-Sided Limit | Exists when lim[x→c⁻] f(x) and lim[x→c⁺] f(x) both exist and are equal; that shared value is lim[x→c] f(x). |
| One-sided Limits | Left-hand limit lim[x→c⁻] f(x) and right-hand limit lim[x→c⁺] f(x); used to determine whether a two-sided limit exists and to classify discontinuities. |
| Indeterminate Form | An expression like 0/0 or ∞/∞ that does not determine the limit's value; signals that algebraic manipulation or another technique is required. |
| Direct Substitution | Evaluating a limit by plugging x = c into f(x); valid when f is continuous at c and produces a defined, non-indeterminate value. |
| Removable Discontinuity | A hole in the graph at x = c where the limit exists but f(c) is undefined or differs from the limit; can be fixed by redefining f(c) to equal the limit. |
| Jump Discontinuity | A discontinuity where both one-sided limits exist but are unequal, causing the graph to jump between two distinct values at x = c. |
| Infinite Limit | A limit where f(x) grows without bound as x approaches c; written lim[x→c] f(x) = ±∞ and associated with a vertical asymptote at x = c. |
| Vertical asymptote | A vertical line x = a where lim[x→a] f(x) = ±∞; occurs at zeros of the denominator in a rational function that do not cancel with the numerator. |
| end behavior | What a function's output approaches as x → +∞ or x → -∞; described precisely using limits at infinity and represented graphically by horizontal asymptotes. |
| Continuous | A function is continuous at x = c when f(c) exists, lim[x→c] f(x) exists, and lim[x→c] f(x) = f(c); no holes, jumps, or asymptotes at that point. |
| Average Rate of Change | (f(a+h) - f(a))/h over an interval; equals the slope of the secant line and approaches the instantaneous rate of change as h → 0. |
| Instantaneous Rate of Change | The limit of the average rate of change as the interval width approaches zero; equals the slope of the tangent line at a point and motivates the definition of the derivative. |
| piecewise-defined function | A function defined by different expressions on different intervals; continuity at boundary points requires the expressions to agree in value at the boundary. |
lim[x→c] f(x) and f(c) are different objects. A function can have a limit at c even if f(c) is undefined or has a different value. Always evaluate them separately before checking continuity.
The form 0/0 is indeterminate, not a final answer. It signals that algebraic manipulation (factoring, conjugate, trig identity) is needed. Only write DNE after confirming the one-sided limits differ or the function is unbounded or oscillatory.
The left-hand and right-hand infinite limits at a vertical asymptote can have opposite signs. For example, lim[x→0⁻] 1/x = -∞ while lim[x→0⁺] 1/x = +∞. Always analyze each side separately.
The IVT requires f to be continuous on the entire closed interval [a, b]. A single jump or infinite discontinuity inside the interval can make the conclusion invalid. State continuity explicitly before using the theorem.
If a common factor cancels in a rational function, the result is a removable discontinuity (hole), not a vertical asymptote. A vertical asymptote requires a zero in the denominator that does not cancel.
AP Calculus questions frequently ask you to justify whether a limit exists, whether a function is continuous at a point, or whether the IVT applies. Partial credit depends on stating conditions explicitly: name the type of discontinuity, identify which continuity condition fails, or confirm all IVT hypotheses before stating the conclusion. Unsupported answers receive little credit even when the final value is correct.
Exam questions present functions as graphs with open and filled circles, as tables with values approaching a point, or as piecewise formulas. You may need to determine a limit, classify a discontinuity, or find a parameter value that ensures continuity from any of these forms. Practice moving between representations so the format of the question does not slow you down.
The limit definition of the derivative lim[h→0] (f(a+h) - f(a))/h in Unit 2 is a direct application of Unit 1 limit skills. Recognizing indeterminate forms, applying algebraic manipulation, and interpreting the result as an instantaneous rate of change all trace back to Unit 1. Exam questions in later units sometimes test whether you can identify a limit expression as a derivative, requiring fluency with both the limit and derivative concepts simultaneously.
Use this list to confirm you can handle every major skill before the exam.
Express one-sided and two-sided limits using correct notation, and interpret what lim[x→c] f(x) = R means graphically, numerically, and analytically.
Apply direct substitution, limit laws, algebraic manipulation (factoring, conjugate), and the Squeeze Theorem. Know which method to choose based on the form of the expression.
Identify removable, jump, and infinite discontinuities from graphs, tables, and formulas. State which of the three continuity conditions fails for each type.
Verify continuity on an interval by checking domain and boundary points. Solve for parameter values that make piecewise functions continuous at their boundaries.
Use infinite limits to locate vertical asymptotes with correct sign analysis. Use limits at infinity and degree comparison to find horizontal asymptotes and describe end behavior.
State all IVT conditions explicitly (continuity on [a, b], target value between f(a) and f(b)), state the conclusion (existence of c), and recognize when the theorem does not apply.
Start with the conceptual motivation in Topic 1.1, then practice reading limits from graphs (1.3) and tables (1.4). Write out limit notation for each example and check that your left-hand and right-hand estimates agree before claiming a two-sided limit exists.
Work through limit laws (1.5), then practice factoring, conjugate rationalization, and trig form substitution (1.6-1.7). Memorize lim[x→0] sin(x)/x = 1 and lim[x→0] (1 - cos x)/x = 0 from the Squeeze Theorem (1.8). Use Topic 1.9 to connect all three representations for the same limit.
Write out the three-condition definition of continuity at a point and practice applying it to rational, piecewise, and graphically defined functions. For each discontinuity you find, name the type and identify which condition fails.
List the standard function families and their domains. Then practice piecewise continuity problems where you solve for an unknown constant by setting boundary expressions equal. Confirm your answer satisfies all three continuity conditions.
Practice sign analysis near vertical asymptotes and degree comparison for horizontal asymptotes. Finish by writing out complete IVT justifications: state the interval, confirm continuity, identify f(a) and f(b), name the target value d, and state the conclusion.
Open the individual guides for Unit 1 when you want a closer review of one topic.
Use AP-style practice after you review the notes so you can check what you understand.
Practice free-response reasoning and compare your answer with scoring guidance.
Watch past review streams filtered to Unit 1 when you want a video walkthrough.
Open the Fiveable one-page unit review, then explore visual cheatsheets for a quick refresher.
Estimate your broader AP score goal after you review the course and exam format.
AP Calc Unit 1 covers 16 topics built around limits and continuity. You'll work through defining and estimating limits from graphs and tables, using algebraic properties and manipulation to evaluate limits, the Squeeze Theorem, types of discontinuities, continuity at a point and over an interval, removing discontinuities, infinite limits, limits at infinity, vertical and horizontal asymptotes, and the Intermediate Value Theorem (IVT). Here's a quick breakdown of the major clusters: - **Limits foundations:** Introducing Calculus, Defining Limits and Using Limit Notation, Estimating Limit Values from Graphs and Tables - **Evaluating limits:** Algebraic Properties of Limits, Algebraic Manipulation, Selecting Procedures, the Squeeze Theorem, Connecting Multiple Representations - **Continuity:** Types of Discontinuities, Continuity at a Point, Continuity over an Interval, Removing Discontinuities - **Asymptotes and IVT:** Infinite Limits and Vertical Asymptotes, Limits at Infinity and Horizontal Asymptotes, the Intermediate Value Theorem See the full topic list at AP Calc Unit 1.
Unit 1 makes up 10-12% of the AP Calc exam, so it's a meaningful but not dominant chunk of your score. The unit covers limits and continuity, including evaluating limits algebraically, identifying discontinuities, connecting limits to asymptotes, and applying the Intermediate Value Theorem. Expect several multiple-choice questions drawn directly from these concepts on exam day.
The AP Calc Unit 1 progress check in AP Classroom includes both MCQ and FRQ parts that test limits and continuity. The MCQ section pulls from topics like estimating limits from graphs and tables, using algebraic manipulation and the Squeeze Theorem, and identifying types of discontinuities. The FRQ part typically asks you to confirm continuity at a point, remove a discontinuity, or apply the Intermediate Value Theorem with written justification. For the progress check, focus especially on: - Evaluating limits using algebraic properties and manipulation (Topics 1.5-1.7) - Identifying and classifying discontinuities (Topics 1.10-1.13) - Connecting limits at infinity to horizontal asymptotes (Topic 1.15) - Applying the IVT with a complete written argument (Topic 1.16) Practice with matched questions at AP Calc Unit 1 before your progress check deadline.
AP Calc Unit 1 FRQs most often ask you to justify continuity at a point, remove a discontinuity by defining or redefining a function value, or apply the Intermediate Value Theorem to guarantee a solution exists on an interval. To practice, work through problems that require written justification, not just a numerical answer, since College Board awards points specifically for the reasoning you write down. A solid practice routine for Unit 1 FRQs: 1. Review the definitions for continuity (Topics 1.11-1.12) so you can write them precisely. 2. Practice IVT problems where you state all three conditions explicitly (Topic 1.16). 3. Work through removing discontinuities by identifying the limit and comparing it to the function value (Topic 1.13). 4. After each attempt, check whether your written justification matches what the scoring guideline expects. Find practice problems and worked examples at AP Calc Unit 1.
The best place to find AP Calc Unit 1 practice questions, including multiple-choice and progress-check style problems, is AP Calc Unit 1. You'll find MCQ practice covering limit estimation from graphs and tables, algebraic limit evaluation, continuity, asymptotes, and the IVT. For a practice-test feel, work through questions from each topic cluster in order so you build the skills progressively before attempting a full mixed set.
Start AP Calc Unit 1 by building a solid understanding of what a limit actually means before touching any algebra. Once the concept clicks, work through the evaluation techniques in order: direct substitution, factoring and simplifying, rationalization, and the Squeeze Theorem. Then shift to continuity, where you'll connect limit skills to classifying and removing discontinuities. A practical study plan: 1. **Understand the concept first.** Read through Topic 1.1 and 1.2 to get comfortable with limit notation and the idea of approaching a value. 2. **Practice estimation.** Use graphs (Topic 1.3) and tables (Topic 1.4) to build intuition before jumping to algebra. 3. **Work through algebraic techniques in sequence.** Topics 1.5-1.8 build on each other, so don't skip ahead. 4. **Memorize the continuity definition.** You need to state it precisely for FRQs: the limit exists, the function is defined, and they're equal. 5. **Nail the IVT.** Topic 1.16 shows up on FRQs regularly. Practice writing out all three conditions every time. 6. **Review asymptotes last.** Topics 1.14-1.15 connect limits to behavior you already know from precalc, so they tend to click quickly. All 16 topics with practice are at AP Calc Unit 1.