Limits and Continuity — AP Calculus AB Unit Overview

For: AP Calculus AB candidates sitting AP Calculus AB.

Covers: All 16 sub-topics of Unit 1: limit definition, estimation, algebraic limit techniques, Squeeze Theorem, discontinuity classification, continuity rules, removable discontinuities, limit-asymptote connections, and applications of the Intermediate Value Theorem (IVT).

You should already know: Basic function algebra including factoring and rational expressions, graphing of common function families, and function notation conventions.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the AP Calculus AB style for educational use. They are not reproductions of past College Board / Cambridge / IB papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official mark schemes for grading conventions.

1. Why This Unit Matters

Limits and Continuity is the foundational first unit of AP Calculus AB, and it makes up 10-12% of your total exam score per the official College Board Course and Exam Description (CED). This unit appears in both multiple-choice (MCQ) and free-response (FRQ) sections, both as standalone questions and as a hidden prerequisite for nearly every other question on the exam.

This unit answers the core question that launched calculus: how can we analyze change that happens over infinitely small intervals, when all we know how to do is calculate change over finite intervals? Every major concept that follows in AP Calculus AB is built on limits: the derivative is defined as a limit of difference quotients, the definite integral is defined as a limit of Riemann sums, and all curve-sketching and optimization techniques rely on understanding continuity and asymptotic behavior. Even problems that don’t explicitly mention “limits” or “continuity” require implicit mastery of this unit. For example, you must confirm continuity to apply major theorems later in the course like the Mean Value Theorem or Fundamental Theorem of Calculus. Without a solid foundation in this unit, you will struggle to interpret or justify results in every subsequent unit.

2. Unit Concept Map

The unit builds incrementally from intuition to formal definition, calculation, and application, following this logical sequence that connects all 16 sub-topics:

1. The unit opens with the motivating question Can change occur at an instant?, which frames why we need the new tool of limits instead of just basic algebra.
2. Next, we formalize defining limits and using limit notation, establishing the convention for two-sided and one-sided limits.
3. We build intuition by estimating limit values from graphs and estimating limit values from tables, learning how limits relate to these common function representations.
4. We move from estimation to exact calculation: first learning determining limits using algebraic properties of limits (sum, product, quotient rules), then determining limits using algebraic manipulation to resolve indeterminate forms.
5. Next, we develop fluency by learning selecting procedures for determining limits, then study the more advanced Squeeze theorem for tricky indeterminate forms involving bounded oscillations.
6. We consolidate understanding by connecting multiple representations of limits (graphical, tabular, algebraic) to each other.
7. We then turn to continuity, starting with exploring types of discontinuities, then formalizing defining continuity at a point, extending that to confirming continuity over an interval, and learning how to fix gaps with removing discontinuities.
8. Finally, we connect limits to key graphical features: connecting infinite limits and vertical asymptotes and connecting limits at infinity and horizontal asymptotes, then end with the unit’s key applied theorem: Working with the Intermediate Value Theorem (IVT).

3. A Guided Tour Through a Typical Exam Problem

To see how the unit’s sub-topics connect to solve a single exam-style problem, consider this common multi-part question that draws on 3 core unit concepts:

Let $f(x)=\frac{x^2-9}{x-3}$ for $x\ne 3$. Answer the following: (1) Find ${\lim }_{x\to 3}f(x)$, (2) What type of discontinuity does $f$ have at $x=3$, (3) Can we redefine $f(3)$ to make $f$ continuous at $x=3$, and if so, what value should we use?

Step 1: Apply core limit calculation techniques

First, we use the sub-topic determining limits using algebraic manipulation to solve for the limit. Direct substitution of $x=3$ gives the indeterminate form $\frac{0}{0}$, so we factor the difference of squares in the numerator: $$x^2-9=(x-3)(x+3)$$ We can cancel $(x-3)$ because for a limit as $x\to 3$, $x$ is never equal to 3, so the factor is non-zero and cancellation is valid. This leaves us with ${\lim }_{x\to 3}(x+3)=6$.

Step 2: Classify the discontinuity

Next, we apply the sub-topics exploring types of discontinuities and defining continuity at a point. For a function to be continuous at $x=a$, three conditions must hold: (1) $f(a)$ is defined, (2) ${\lim }_{x\to a}f(x)$ exists, (3) the limit equals $f(a)$. Here, $f(3)$ is not defined in the original function, but the limit exists, so this is a removable discontinuity (distinct from jump or infinite discontinuities, which require non-existent two-sided limits).

Step 3: Remove the discontinuity

Finally, we apply the sub-topic removing discontinuities. To satisfy all three conditions for continuity, we set $f(3)$ equal to the existing limit: $f(3)=6$. This removes the discontinuity and makes $f$ continuous at $x=3$.

Exam note: This type of multi-part question is extremely common on the AP exam, testing how you connect multiple sub-topics from the unit instead of just recalling a single rule.

4. Common Cross-Cutting Pitfalls (and how to avoid them)

• Wrong move: Automatically equating ${\lim }_{x\to a}f(x)$ to $f(a)$ for any function. Why: Students overgeneralize the direct substitution property, which only holds for continuous functions, so they forget that the limit and the function value are separate concepts. Correct move: Always calculate ${\lim }_{x\to a}f(x)$ and $f(a)$ separately, and only equate them after you confirm $f$ is continuous at $a$.
• Wrong move: Concluding the original function is continuous at $x=a$ after canceling $(x-a)$ to find the limit. Why: Canceling the factor removes the discontinuity algebraically for calculation, but it does not change the original function’s domain. Correct move: After finding the limit, always check whether the original function is defined at $x=a$ and meets all three continuity conditions before concluding continuity.
• Wrong move: Skipping the continuity check when applying the Intermediate Value Theorem. Why: Most textbook and exam problems use continuous functions, so students get in the habit of assuming the hypothesis holds without stating it. AP exam requires you to confirm the hypothesis to earn full points. Correct move: Always explicitly state that $f$ is continuous on the given closed interval before writing any conclusion from IVT.
• Wrong move: Concluding a two-sided limit exists when only one side approaches a finite value. Why: Students often stop calculating after finding one side’s limit, especially if the problem does not explicitly mention one-sided limits. Correct move: For any two-sided limit ${\lim }_{x\to a}f(x)$, always confirm that the left-hand limit equals the right-hand limit before concluding the limit exists.
• Wrong move: Claiming a function has no horizontal asymptote if ${\lim }_{x\to \infty }f(x)$ is infinite. Why: Students mix up the definitions of vertical and horizontal asymptotes, where vertical asymptotes correspond to infinite limits, so they incorrectly assume infinite limits mean no horizontal asymptote. Correct move: Evaluate both ${\lim }_{x\to \infty }f(x)$ and ${\lim }_{x\to -\infty }f(x)$; any finite limit value gives a horizontal asymptote at that $y$-value, even if the other limit is infinite.
• Wrong move: Estimating a limit from a table by only reading the value at $x=a$ instead of checking the trend from both sides. Why: Students confuse the function value at $a$ with the limit as $x$ approaches $a$, even when working with tables. Correct move: When estimating a limit from a table, always check the trend of values approaching $a$ from the left and from the right to confirm they converge to the same value.

5. Quick Check: Do You Know When to Use Which Sub-Topic?

Test your understanding by matching each scenario to the correct unit sub-topic:

1. You need to find ${\lim }_{x\to 0}{x}^2\sin \left(\frac{1}{x}\right)$, where direct substitution gives an undefined, oscillating value.
2. You need to find ${\lim }_{x\to 2}\frac{x^2-3x+2}{x-2}$, which is indeterminate $\frac{0}{0}$ at $x=2$.
3. You need to find the equation of all horizontal asymptotes of a rational function.
4. You need to confirm that a function satisfies the hypothesis for the Mean Value Theorem (coming later in the course) on the interval $[a,b]$.
5. You need to prove that $f(x)=x^3-2x-1$ has at least one root between $x=1$ and $x=2$.

Answers:

1. Squeeze theorem
2. Determining limits using algebraic manipulation
3. Connecting limits at infinity and horizontal asymptotes
4. Confirming continuity over an interval
5. Working with the Intermediate Value Theorem (IVT)

6. All Unit Sub-Topics (See Also)

Below are links to each individual sub-topic study guide in this unit:

• Can change occur at an instant?
• Defining limits and using limit notation
• Estimating limit values from graphs
• Estimating limit values from tables
• Determining limits using algebraic properties of limits
• Determining limits using algebraic manipulation
• Selecting procedures for determining limits
• Squeeze theorem
• Connecting multiple representations of limits
• Exploring types of discontinuities
• Defining continuity at a point
• Confirming continuity over an interval
• Removing discontinuities
• Connecting infinite limits and vertical asymptotes
• Connecting limits at infinity and horizontal asymptotes
• Working with the Intermediate Value Theorem (IVT)

What's Next

This unit is the foundation for every upcoming unit in AP Calculus AB. Immediately after mastering Limits and Continuity, you will move on to Unit 2: Differentiation: Definition and Fundamental Properties, where you will define the derivative as the limit of a difference quotient. Without understanding how limits work and when they exist, you cannot correctly define or interpret derivatives. This unit also underpins all future work with curve analysis, where you will use limits to find asymptotes and confirm the continuity requirements for theorems like the Mean Value Theorem and Extreme Value Theorem. Follow the links above to study each sub-topic in depth, starting with the motivating opening question of the unit.