# Defining limits and using limit notation

> AP Calculus AB · Unit 1: Limits and Continuity
> Source: https://www.owlsprep.com/study/ap-calculus-ab-u1-defining-limits-and-using-limit/

This module covers intuitive definitions of two-sided and one-sided limits, infinite limits and limits at infinity, standard limit notation, and the key distinction between limit values and function values at a point, aligned to AP Calculus AB requirements.

**Prerequisites:** Basic function notation and evaluation; Algebraic simplification of polynomials and rational functions; Graph reading for discontinuities and asymptotes

## Learning objectives

- Intuitively define what a limit represents
- Use correct notation for two-sided, one-sided, infinite, and limits at infinity
- Distinguish between the limit value of a function at a point and the function value at that point
- Determine if a two-sided limit exists using one-sided limits

## Core Definition of a Limit and Two-Sided Notation

A limit describes the behavior of a function $f(x)$ as $x$ approaches a specific input value $a$, regardless of the actual value of $f(a)$ at that input. This is the foundational concept for all of calculus: every derivative and integral is defined using a limit. The AP exam expects you to translate between verbal descriptions, notation, graphs, and tables of limit behavior.

**Two-Sided Limit** — A two-sided limit describes the value $L$ that $f(x)$ approaches as $x$ approaches $a$ from both the left (values less than $a$) and right (values greater than $a$). $x$ gets arbitrarily close to $a$ but never actually equals $a$.

*Notation:* \lim_{x \to a} f(x) = L

*Example:* A two-sided limit can exist even if $f(a)$ is undefined or different from $L$.

**Worked example:** The function $f(x)$ is defined as $f(x) = \frac{x^2 - 9}{x - 3}$ for $x \neq 3$, 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 limit value.

1. This is an unrestricted two-sided limit with $a=3$, so we use standard two-sided limit notation.
2. $$\lim_{x \to 3} f(x) = L$$
3. Simplify the expression to find $L$ by factoring the numerator:
4. $$x^2 - 9 = (x-3)(x+3)$$
5. For $x \neq 3$, we can cancel the common $(x-3)$ term, leaving $f(x) = x + 3$.
6. As $x$ approaches 3, $x+3$ approaches $3+3=6$, so the full correct notation and value is:
7. $$\lim_{x \to 3} f(x) = 6$$

> **Exam tip:** Never write $f(3) = 6$ for this problem. AP graders deduct points for confusing the limit value with the function value at $a$, even if you calculate the correct limit.

## One-Sided Limits and Existence Rule

One-sided limits describe function behavior when $x$ approaches $a$ from only one direction. AP exams strictly grade correct notation for one-sided limits, and use the connection between one-sided limits and two-sided limit existence for many problems, especially with piecewise functions.

**One-Sided Limits** — Left-hand limits: $x$ approaches $a$ from values less than $a$. Right-hand limits: $x$ approaches $a$ from values greater than $a$. The direction superscript always goes on $a$, not $x$.

*Notation:* Left-hand: $\lim_{x \to a^-} f(x) = L$, Right-hand: $\lim_{x \to a^+} f(x) = L$

*Example:* Commonly used to evaluate limits for piecewise functions at the junction of two pieces.

$$\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$$

**Worked example:** Given the piecewise function $g(x) = \begin{cases} 2x + 1 & x < 2 \\ x^2 - 1 & x \geq 2 \end{cases}$, 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 $x \to 2^-$, use the piece of the function defined for $x < 2$, which is $2x+1$.
2. As $x$ approaches 2 from the left, $2x+1$ approaches $2(2)+1 = 5$, so:
3. $$\lim_{x \to 2^-} g(x) = 5$$
4. For the right-hand limit $x \to 2^+$, use the piece defined for $x \geq 2$, which is $x^2 -1$.
5. As $x$ approaches 2 from the right, $x^2 -1$ approaches $4 - 1 = 3$, so:
6. $$\lim_{x \to 2^+} g(x) = 3$$
7. A two-sided limit only exists if both one-sided limits are equal. Since $5 \neq 3$, $\lim_{x \to 2} g(x)$ does not exist.

> **Exam tip:** Always remember the direction 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.

## Infinite Limits vs Limits at Infinity

Students often confuse these two limit types, which have distinct definitions and notation that are regularly tested on the AP exam. An infinite limit describes unbounded function growth as $x$ approaches a finite $a$, while a limit at infinity describes function behavior as $x$ itself grows without bound.

**Infinite Limit** — Describes a function that grows without bound toward positive infinity or decreases without bound toward negative infinity as $x$ approaches a finite $a$. Since infinity is not a real number, this notation means the limit does not exist as a finite value.

*Notation:* $\lim_{x \to a} f(x) = \pm \infty$

*Example:* Used to describe vertical asymptotes.

**Limit at Infinity** — Describes the value $L$ that a function approaches as $x$ grows without bound toward positive or negative infinity. If $L$ is finite, the limit exists.

*Notation:* $\lim_{x \to \pm \infty} f(x) = L$

*Example:* Used to describe end behavior and 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, $h(x) = \frac{1}{x-1}$ 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:
2. $$\lim_{x \to 1^+} \frac{1}{x-1} = \infty$$
3. For part (b), we have a limit at infinity where $x$ approaches positive infinity, and the function approaches 0. The correct notation is:
4. $$\lim_{x \to \infty} \frac{1}{x-1} = 0$$

> **Exam tip:** If a multiple-choice question asks whether a limit exists when $\lim_{x \to a} f(x) = \infty$, the correct answer is usually that the limit does not exist, as infinity is not a finite real number.

## AP-Style Concept Check

**Check your understanding**

Test your understanding of limit notation and core concepts with these AP-style multiple choice questions.

1. 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) $\lim_{x^- \to -4} g(x) = -7$

   *Why:* Correct. The value $x$ approaches is $-4$, direction from the left requires a negative superscript on $a$, and the superscript goes on $a$, not $x$.

2. Let $k(x) = \begin{cases} \dfrac{x^2 - 2x - 8}{x + 2} & x \neq -2 \\ 3 & x = -2 \end{cases}$. What is $\lim_{x \to -2} k(x)$?

   - A) Does not exist
   - B) $-6$
   - C) $3$
   - D) $-4$

   *Why:* Correct. Factor the numerator: $x^2-2x-8=(x-4)(x+2)$, cancel $(x+2)$ for $x \neq -2$ to get $x-4$, which approaches $-2-4=-6$. The function value $k(-2)=3$ does not affect the limit.

## Common pitfalls

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

## Cheatsheet

| Category | Notation | Notes |
| --- | --- | --- |
| Two-sided finite limit | $\lim_{x \to a} f(x) = L$ | Exists iff both one-sided limits equal $L$; describes approach to finite $a$ from both sides. |
| Left-hand one-sided limit | $\lim_{x \to a^-} f(x) = L$ | Approach from values less than $a$; superscript goes on $a$, not $x$. |
| Right-hand one-sided limit | $\lim_{x \to a^+} f(x) = L$ | Approach from values greater than $a$; superscript goes on $a$, not $x$. |
| Two-sided limit existence rule | $\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-}f(x) = \lim_{x \to a^+}f(x) = L$ | If one-sided limits are not equal, two-sided limit does not exist. |
| Infinite limit (finite $a$) | $\lim_{x \to a} f(x) = \pm \infty$ | Describes unbounded growth/decay; infinity is not a number, so limit does not exist as finite value. |
| Limit at infinity | $\lim_{x \to \pm \infty} f(x) = L$ | Describes end behavior as $x$ grows without bound; if $L$ is finite, the limit exists. |
| Limit vs function value | $\lim_{x \to a} f(x)$ is not necessarily $f(a)$ | Limit depends on behavior near $a$, not at $a$; limit can exist if $f(a)$ is undefined. |
| One-sided limit for piecewise functions | Use the piece defined for the direction of approach | For $x \to a^-$, use the piece valid for $x < a$; for $x \to a^+$, use the piece valid for $x > a$. |

## 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 every concept that follows. Errors in limit notation or understanding the core definition will cascade into mistakes later in the course, including when you write the limit definition of a derivative, a common AP free-response question topic. Immediately after this sub-topic in Unit 1, you will build on this foundation to estimate limits from graphs and tables, calculate limits algebraically, and apply limits to define continuity and find asymptotes. This topic connects directly to every major calculus concept, from derivatives to integrals, so a solid understanding here will make all future topics easier.

- [Estimating limit values from graphs](https://www.owlsprep.com/study/ap-calculus-ab-u1-estimating-limit-values-from-graphs/)
- [Estimating limit values from tables](https://www.owlsprep.com/study/ap-calculus-ab-u1-estimating-limit-values-from-tables/)
- [Determining limits using algebraic properties of limits](https://www.owlsprep.com/study/ap-calculus-ab-u1-determining-limits-using-algebraic-properties/)

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