Defining limits and using limit notation — AP Calculus AB Study Guide

For: AP Calculus AB candidates sitting AP Calculus AB.

Covers: Intuitive definition of two-sided and one-sided limits, infinite limits and limits at infinity, standard limit notation conventions, and distinguishing between limit values and function values at a point.

You should already know: Basic function notation and evaluation. Algebraic simplification of polynomials and rational functions. Graph reading for discontinuities and asymptotes.

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. What Is Defining limits and using limit notation?

A limit describes the behavior of a function $f(x)$ as $x$ approaches a specific input value, regardless of the actual value of $f$ at that input. This is the foundational concept for all of calculus: every derivative and integral is defined using a limit, so mastery of notation and definition is non-negotiable for the entire course. According to the AP Calculus AB Course and Exam Description (CED), Unit 1: Limits and Continuity accounts for 10-12% of the total exam score, and this opening topic is tested in both multiple-choice (MCQ) and free-response (FRQ) sections. You will see stand-alone questions testing your ability to read or write limit notation, interpret limits from graphs or tables, and distinguish between ${\lim }_{x\to a}f(x)$ and $f(a)$. You will also need to correctly write limit notation for derivatives and areas later in the course, so errors here will cascade into later problems. The AP exam expects you to translate between verbal descriptions of function behavior, limit notation, graphs, and tables, which we break down into core sub-concepts below.

2. Two-Sided Limits and Standard Notation

A two-sided limit describes the value the function approaches as $x$ approaches $a$ from both the left (values less than $a$) and the right (values greater than $a$). The standard notation for a two-sided limit is: $$\underset{x\to a}{\lim }f(x)=L$$ where $x\to a$ means $x$ gets arbitrarily close to $a$, but never actually equals $a$, and $L$ is the finite value the function approaches. This is the most common limit you will work with on the AP exam.

The key distinction AP examiners repeatedly test is that the limit is about the trend of the function near $a$, not the value of the function at $a$. A function can have a perfectly valid limit at $a$ even if $f(a)$ is undefined, or if $f(a)$ is a different value than the limit. This is most commonly seen with rational functions that have removable discontinuities (holes).

Worked Example

The function $f(x)$ is defined as $f(x)=\frac{x^2-9}{x-3}$ for $x\ne 3$, and $f(3)=10$. Write the correct limit notation for the value $f(x)$ approaches as $x$ gets arbitrarily close to 3, then find the value of the limit.

1. This is a two-sided limit, as there is no restriction on approaching 3 from only one direction, so we use standard two-sided limit notation with $a=3$.
2. The base notation is ${\lim }_{x\to 3}f(x)=L$, where $L$ is the limit value.
3. Simplify the expression to find $L$: factor the numerator: $x^2-9=(x-3)(x+3)$. For $x\ne 3$, we can cancel the common $(x-3)$ term, leaving $f(x)=x+3$.
4. As $x$ approaches 3, $x+3$ approaches $3+3=6$, so the full correct notation and value is ${\lim }_{x\to 3}f(x)=6$.

Exam tip: Never write $f(3)=6$ for this type of problem. AP graders will deduct points if you confuse the function value at $a$ with the limit as $x$ approaches $a$, even if you calculate the correct limit value.

3. One-Sided Limits

One-sided limits are limits where $x$ approaches $a$ from only one direction, either the left (values less than $a$) or the right (values greater than $a$). The notation conventions are strictly graded on the AP exam: a left-hand limit (approach from the left) is written ${\lim }_{x\to a^{-}}f(x)=L$, where the negative superscript is on $a$, not $x$. A right-hand limit (approach from the right) is written ${\lim }_{x\to a^{+}}f(x)=L$, with the positive superscript on $a$.

A core rule for two-sided limits relies on one-sided limits: the two-sided limit ${\lim }_{x\to a}f(x)$ exists and equals $L$ if and only if both one-sided limits exist and equal $L$. In other words: $$\underset{x\to a}{\lim }f(x)=L⟺\underset{x\to a^{-}}{\lim }f(x)=\underset{x\to a^{+}}{\lim }f(x)=L$$ This rule is the foundation for working with piecewise functions, which are extremely common on AP exam questions testing one-sided limits.

Worked Example

Given the piecewise function $g(x)=\{\begin{array}{ll}2x+1& x<2\\ x^2-1& x\ge 2\end{array}$, find ${\lim }_{x\to 2^{-}}g(x)$ and ${\lim }_{x\to 2^{+}}g(x)$, then state whether ${\lim }_{x\to 2}g(x)$ exists.

1. For the left-hand limit $x\to 2^{-}$, we use the piece of the function defined for all $x<2$, which is $2x+1$.
2. Calculate the trend: as $x$ approaches 2 from the left, $2x+1$ approaches $2(2)+1=5$, so ${\lim }_{x\to 2^{-}}g(x)=5$.
3. For the right-hand limit $x\to 2^{+}$, we use the piece defined for all $x\ge 2$, which is $x^2-1$. As $x$ approaches 2 from the right, $x^2-1$ approaches $4-1=3$, so ${\lim }_{x\to 2^{+}}g(x)=3$.
4. The two-sided limit only exists if both one-sided limits are equal. Since $5\ne 3$, ${\lim }_{x\to 2}g(x)$ does not exist.

Exam tip: Always remember the superscript goes on $a$, not $x$: the notation ${\lim }_{x^{-}\to 2}g(x)$ is incorrect and will be marked wrong on the AP exam.

4. Infinite Limits and Limits at Infinity

Students often confuse these two related limit types, but they have distinct definitions and notation that you must master for the exam. An infinite limit is a limit where the function grows without bound (toward positive or negative infinity) as $x$ approaches a finite $a$. A limit at infinity is a limit where $x$ itself grows without bound (toward positive or negative infinity), and the function approaches a finite value $L$.

Notation for infinite limits: if $f(x)$ increases without bound as $x\to a$, we write ${\lim }_{x\to a}f(x)=\infty$. If it decreases without bound, we write ${\lim }_{x\to a}f(x)=-\infty$. Important: saying the limit equals infinity does not mean the limit exists — infinity is not a real number, this notation just describes the trend of unbounded growth. For limits at infinity, the notation is ${\lim }_{x\to \infty }f(x)=L$ (x grows without bound positive) or ${\lim }_{x\to -\infty }f(x)=L$ (x grows without bound negative). This notation is used to describe end behavior of functions and find horizontal asymptotes.

Worked Example

Write the correct limit notation for each verbal description: (a) As $x$ approaches 1 from the right, the function $h(x)=\frac{1}{x-1}$ grows without bound toward positive infinity. (b) As $x$ grows larger and larger without bound, the function $h(x)=\frac{1}{x-1}$ approaches 0.

1. For part (a), we have a one-sided infinite limit approaching 1 from the right. The direction requires a positive superscript on $a=1$, and the result is positive infinity. The correct notation is ${\lim }_{x\to 1^{+}}\frac{1}{x-1}=\infty$.
2. For part (b), we have a limit at infinity where $x$ approaches positive infinity and the function approaches 0. The value being approached by $x$ is positive infinity, and the limit value is 0.
3. The correct notation for part (b) is ${\lim }_{x\to \infty }\frac{1}{x-1}=0$, which matches the verbal description exactly.

Exam tip: If a multiple-choice question asks whether a limit exists when the limit equals infinity, the correct answer is usually "the limit does not exist" — writing $\lim =\infty$ is just notation for unbounded growth, not an existing finite limit.

5. Common Pitfalls (and how to avoid them)

• Wrong move: Writing ${\lim }_{x^{-}\to 2}f(x)$ instead of ${\lim }_{x\to 2^{-}}f(x)$ for a left-hand limit. Why: Students confuse which variable the direction applies to; the direction is relative to $a$, not $x$. Correct move: Always place the positive/negative superscript on the value $a$ that $x$ is approaching.
• Wrong move: Stating that ${\lim }_{x\to 3}f(x)=10$ to match the function value $f(3)=10$, even though the function approaches 6 near $x=3$. Why: Students confuse the function's value at the point with the limit's value near the point. Correct move: First ask "what value does $f$ approach as $x$ gets close to $a$", then check what $f(a)$ is separately — never assume they are equal.
• Wrong move: Claiming that ${\lim }_{x\to a}f(x)$ does not exist because $f(a)$ is undefined. Why: Students associate function value with limit value, so they assume no function value means no limit. Correct move: If asked if the limit exists, check the trend of $f(x)$ for $x$ near $a$, not at $a$; a limit can exist even if $f(a)$ is undefined.
• Wrong move: Writing ${\lim }_{x\to \infty }f(x)=DNE$ when ${\lim }_{x\to \infty }f(x)=5$. Why: Students confuse "infinity in the notation" with "infinite limit"; they think any limit with infinity in the notation does not exist. Correct move: Only label a limit as DNE if there is no finite $L$ that the function approaches; if $x$ approaches infinity and $f(x)$ approaches 5, the limit exists and equals 5.
• Wrong move: Stating that ${\lim }_{x\to 2}f(x)=3$ when ${\lim }_{x\to 2^{-}}f(x)=5$ and ${\lim }_{x\to 2^{+}}f(x)=3$. Why: Students only check the right-hand limit or the limit from the side matching the function definition at $a$, instead of both sides. Correct move: To find a two-sided limit, always calculate both one-sided limits first and confirm they are equal before reporting the two-sided limit.
• Wrong move: Writing ${\lim }_{x\to a}f(x)=\infty$ and claiming this means the limit exists. Why: Students see the equals sign and infinity written, so they think that means the limit exists. Correct move: Remember that infinity is not a real number; writing $\lim =\infty$ is just a notation for unbounded growth, and the limit still does not exist as a finite value.

6. Practice Questions (AP Calculus AB Style)

Question 1 (Multiple Choice)

Which of the following is the correct limit notation for the statement: "The value that $g(x)$ approaches as $x$ approaches $-4$ from the left is $-7$." A) ${\lim }_{x\to -4^{+}}g(x)=-7$ B) ${\lim }_{x\to -7^{-}}g(x)=-4$ C) ${\lim }_{x\to -4^{-}}g(x)=-7$ D) ${\lim }_{x^{-}\to -4}g(x)=-7$

Worked Solution: First, parse the statement: the value $x$ is approaching is $-4$, so $a=-4$, which eliminates option B that incorrectly sets $a=-7$. The direction is approaching from the left, which requires a negative superscript on $a$, eliminating option A that uses a positive superscript. The superscript always goes on $a$, not $x$, so D is incorrectly formatted. Only option C matches the statement with correct notation. Correct answer: C.

Question 2 (Free Response)

Let $k(x)$ be defined as $$k(x)=\{\begin{array}{ll}\frac{x^2-2x-8}{x+2}& x\ne -2\\ 3& x=-2\end{array}$$ (a) Write the notation for the two-sided limit of $k(x)$ as $x$ approaches $-2$. (b) Calculate ${\lim }_{x\to -2^{-}}k(x)$ and ${\lim }_{x\to -2^{+}}k(x)$. (c) State whether ${\lim }_{x\to -2}k(x)$ exists, and give its value if it exists. Then compare the limit value to $k(-2)$.

Worked Solution: (a) The standard notation for the two-sided limit as $x$ approaches $-2$ is: $$\underset{x\to -2}{\lim }k(x)$$ (b) For both one-sided limits, $x\ne -2$, so we use the rational expression. Factor the numerator: $x^2-2x-8=(x-4)(x+2)$. Cancel $(x+2)$ (valid for $x\ne -2$) to get $k(x)=x-4$. Evaluate the left-hand limit: ${\lim }_{x\to -2^{-}}(x-4)=-2-4=-6$. Evaluate the right-hand limit: ${\lim }_{x\to -2^{+}}(x-4)=-2-4=-6$. So ${\lim }_{x\to -2^{-}}k(x)={\lim }_{x\to -2^{+}}k(x)=-6$. (c) Since both one-sided limits exist and are equal to $-6$, the two-sided limit ${\lim }_{x\to -2}k(x)$ exists and equals $-6$. By definition, $k(-2)=3$, so the limit value $(-6)$ is not equal to the function value at $x=-2$ ($3$).

Question 3 (Application / Real-World Style)

A cylindrical tank draining through a small hole has height of water given by $h(t)=(2-0.1t{)}^2$ centimeters after $t$ minutes, for $0\le t<20$. The tank is defined as empty at $t=20$. What is the meaning of ${\lim }_{t\to 20^{-}}h(t)$ in context, and what is the value of this limit?

Worked Solution: The notation ${\lim }_{t\to 20^{-}}$ means we measure the height of water as $t$ approaches 20 minutes from the left, that is, for times just before 20 minutes. Substitute to find the limit: as $t$ approaches 20 from the left, $(2-0.1t)$ approaches $2-0.1(20)=0$, so $(2-0.1t{)}^2$ approaches $0^2=0$. We get ${\lim }_{t\to 20^{-}}h(t)=0$ centimeters. In context, this means that just before 20 minutes pass, the height of water in the tank approaches 0 cm, so the tank is nearly empty right at the 20 minute mark, matching expectations.

7. Quick Reference Cheatsheet

8. What's Next

This topic is the absolute foundation of all of calculus, so mastering notation and the core idea that limits describe behavior near a point, not at a point, is critical for everything that comes next. Immediately after this in Unit 1, you will learn to estimate limits from graphs and tables, then calculate limits algebraically, and apply limits to find asymptotes and define continuity. Without correctly understanding limit notation and the core definition of a limit, you will not be able to correctly write the limit definition of the derivative (a common AP FRQ topic) or interpret limits in applied problems across the course. This topic feeds into every major concept in calculus, from derivatives to integrals.

Estimating limits from graphs and tables Algebraic techniques for calculating limits Continuity as a property of functions Defining the derivative with limits