# Defining limits and using limit notation

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

This sub-topic covers intuitive and formal epsilon-delta definitions of one-sided, two-sided, infinite, and limits at infinity, standard notation conventions, and interpretation in graphical and contextual problems for AP Calculus BC.

**Prerequisites:** Function notation and domain/range for common function types; Basic graphing of functions including asymptotes; Algebraic manipulation of polynomials and rational expressions

## Learning objectives

- Interpret standard limit notation for one-sided, two-sided, infinite, and infinite limits
- Apply the two-sided limit existence theorem to piecewise and absolute value functions
- Distinguish between infinite limits at finite points and limits at infinity
- Interpret the formal epsilon-delta definition of a limit
- Avoid common exam pitfalls with limit notation and definitions

## Core Definition of Limits and Standard Notation

Limits are the foundational building block of all calculus, describing the behavior of a function $f(x)$ as $x$ approaches a specific input, regardless of the value of $f$ at that input. This topic is embedded across all AP Calculus BC questions, as every core calculus concept (derivatives, integrals) is defined using limits.

**Limit** — The value $L$ that $f(x)$ approaches arbitrarily closely as $x$ gets arbitrarily close (but not equal) to $a$. Unlike function evaluation, limits describe approaching, not being at the point $a$.

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

> **note**
>
> Pronounce $\lim_{x \to a} f(x) = L$ as *the limit of f of x as x approaches a equals L*.

## One-Sided and Two-Sided Limits

A one-sided limit describes the behavior of $f(x)$ as $x$ approaches $a$ from only one side of the number line. Left-hand (from values less than $a$) and right-hand (from values greater than $a$) limits have distinct notation:

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

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

The core existence theorem for two-sided limits states that a two-sided limit $\lim_{x \to a} f(x)$ exists if and only if both one-sided limits exist and are equal. Formally:

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

**Worked example:** Given the piecewise function $f(x) = \begin{cases} 2x + 1 & x < 3 \\ x^2 - 2x & x > 3 \end{cases}$, find $\lim_{x \to 3} f(x)$ if it exists.

1. Calculate the left-hand limit as $x$ approaches 3 from the left. For $x < 3$, $f(x) = 2x + 1$:
2. $$\lim_{x \to 3^-} f(x) = 2(3) + 1 = 7$$
3. Next, calculate the right-hand limit as $x$ approaches 3 from the right. For $x > 3$, $f(x) = x^2 - 2x$:
4. $$\lim_{x \to 3^+} f(x) = (3)^2 - 2(3) = 9 - 6 = 3$$
5. Compare the two one-sided limits: $7 \neq 3$. By the existence theorem for two-sided limits, the two-sided limit $\lim_{x \to 3} f(x)$ does not exist.

> **Exam tip:** On multiple-choice questions asking if a two-sided limit exists for a piecewise function, always check both one-sided limits explicitly—don’t assume they match just because the function is defined at $x=a$ on both pieces.

## Infinite Limits and Vertical Asymptotes

Infinite limits describe the behavior of $f(x)$ as $x$ approaches a finite value $a$, when $f(x)$ grows without bound (toward positive or negative infinity) instead of approaching a finite value $L$. Notation for infinite limits is:

$$\lim_{x \to a} f(x) = \infty \quad \text{or} \quad \lim_{x \to a} f(x) = -\infty$$

A critical point to remember: this notation only describes the unbounded behavior of $f(x)$ near $a$. It does not mean the limit exists as a finite real number. When a function has an infinite limit at $x=a$, it has a vertical asymptote at $x=a$. Infinite limits almost always occur at points where the denominator of a rational function is zero and the numerator is non-zero.

**Worked example:** Find $\lim_{x \to 2^-} \frac{3x + 1}{x - 2}$, and state whether the function has a vertical asymptote at $x=2$.

1. First evaluate the numerator at $x=2$: $3(2) + 1 = 7$, which is non-zero, so we expect an infinite limit at this point.
2. For the left-hand limit $x \to 2^-$, all $x$ near 2 are less than 2, so $x - 2$ is a very small negative number.
3. We have a positive numerator (7) divided by a small negative number, which produces a large negative result. Thus:
4. $$\lim_{x \to 2^-} \frac{3x + 1}{x - 2} = -\infty$$
5. Since an infinite limit exists at $x=2$, the function has a vertical asymptote at $x=2$.

> **Exam tip:** When writing infinite limits on FRQ, saying "the limit equals infinity" is acceptable notation for describing unbounded behavior, but you must answer that the limit does not exist if explicitly asked whether a finite limit exists.

## Formal Epsilon-Delta Definition of a Limit

The intuitive definition of a limit ("$f(x)$ gets close to $L$ when $x$ gets close to $a$") is imprecise, so the formal epsilon-delta definition gives a rigorous way to prove a limit equals a given value $L$.

**Epsilon-Delta Limit Definition** — For every $\varepsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - a| < \delta$, then $|f(x) - L| < \varepsilon$. $\varepsilon$ is the maximum allowed error between $f(x)$ and $L$, and $\delta$ is how close $x$ must be to $a$ (excluding $x=a$ itself) to satisfy the error bound.

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

On the AP Calculus BC exam, you will rarely be asked to write a full epsilon-delta proof, but you will often be asked to interpret the definition or find $\delta$ for a given $\varepsilon$ on multiple-choice questions.

**Worked example:** For $\lim_{x \to 4} (3x - 5) = 7$, find $\delta$ that satisfies the epsilon-delta condition for $\varepsilon = 0.1$.

1. The definition requires that $|(3x - 5) - 7| < 0.1$ whenever $0 < |x - 4| < \delta$.
2. Simplify the left-hand inequality:
3. $$|3x - 12| = 3|x - 4| < 0.1$$
4. Divide both sides by 3 to isolate $|x - 4|$:
5. $$|x - 4| < \frac{0.1}{3} \approx 0.0333$$
6. Thus $\delta = 0.0333$ (or any smaller positive number) satisfies the condition for $\varepsilon = 0.1$.

> **Exam tip:** When finding $\delta$ for a linear function $f(x) = mx + b$, $\delta$ will always equal $\frac{\varepsilon}{|m|}$—use this shortcut to save time on MCQ.

## AP-Style Concept Check

**Check your understanding**

Test your understanding of limit definitions and notation:

1. Given $g(x) = \frac{|x - 4|}{x - 4}$, what is $\lim_{x \to 4} g(x)$?

   - 0
   - 1
   - -1
   - The limit does not exist

   *Why:* Correct. Rewrite the absolute value as a piecewise function: $\lim_{x \to 4^-} g(x) = -1$ and $\lim_{x \to 4^+} g(x) = 1$, so the two-sided limit does not exist.

2. Let $h(x) = \begin{cases} x^2 + k & x \leq 2 \\ 3x + 2k & x > 2 \end{cases}$. For what value of $k$ does $\lim_{x \to 2} h(x)$ exist?

   - k=-2
   - k=2
   - k=-1
   - k=1

   *Why:* Correct. Set the one-sided limits equal: $4 + k = 6 + 2k \implies k=-2$.

3. An ecologist models deer population as $P(t) = \frac{1200t}{t + 5}$ for $t \geq 0$, where carrying capacity is 1200. What is the correct limit notation for long-term population behavior?

   - $\lim_{t \to 1200} P(t) = \infty$
   - $\lim_{t \to \infty} P(t) = 1200$
   - $\lim_{P \to 1200} t = \infty$
   - $\lim_{t \to 1200} P(t) = t$

   *Why:* Correct. Long-term behavior means $t$ (time) approaches infinity, and the limit approaches the carrying capacity of 1200.

## Common pitfalls

- **Wrong:** After calculating only one one-sided limit for a piecewise function at the boundary, concluding the two-sided limit equals that value.
  - Why it fails: Students rush to answer and forget that two-sided limits require matching one-sided limits by definition.
  - Correct: Always explicitly calculate left-hand and right-hand limits, then compare them before concluding whether the two-sided limit exists.
- **Wrong:** Stating that $\lim_{x \to a} f(x) = \infty$ means the limit exists.
  - Why it fails: Students confuse notation for describing unbounded behavior with the definition of an existing finite limit.
  - Correct: When asked if a finite limit exists, if the limit is infinite, answer that the limit does not exist, and note the notation only describes behavior.
- **Wrong:** Evaluating $f(a)$ to find $\lim_{x \to a} f(x)$, and concluding the limit equals $f(a)$.
  - Why it fails: Students confuse function evaluation with limit behavior, especially when working with continuous functions.
  - Correct: Always check one-sided limits first, even if $f(a)$ is defined—discontinuities can cause the limit to differ from $f(a)$ or not exist at all.
- **Wrong:** In epsilon-delta notation, dropping the $0 < |x - a|$ condition and writing only $|x - a| < \delta$.
  - Why it fails: Students forget that the limit does not depend on the value of $f$ at $x=a$, only on values near $a$.
  - Correct: Always include the $0 <$ inequality in epsilon-delta statements to exclude $x=a$ itself.
- **Wrong:** Confusing $\lim_{x \to \infty} f(x)$ (limit at infinity) with $\lim_{x \to a} f(x) = \infty$ (infinite limit at finite $a$).
  - Why it fails: Both involve infinity in the notation, leading to mix-ups about the behavior being described.
  - Correct: Always check where the infinity is placed: infinity under the limit sign means $x$ grows without bound; infinity after the equals sign means $f(x)$ grows at a finite $a$.
- **Wrong:** For absolute value functions, using the same expression for left and right limits at the critical point.
  - Why it fails: Students forget $|x - a|$ is a piecewise function that changes definition at $x=a$.
  - Correct: Rewrite $|x - a|$ into its piecewise form before calculating one-sided limits at $x=a$.

## Cheatsheet

| Category | Formula/Notation | Notes |
| --- | --- | --- |
| Two-sided limit notation | $\lim_{x \to a} f(x) = L$ | Describes $f(x)$ approaching $L$ as $x$ approaches $a$ (excludes $x=a$) |
| Left-hand one-sided limit | $\lim_{x \to a^-} f(x) = L$ | $x$ approaches $a$ from values *less than* $a$ |
| Right-hand one-sided limit | $\lim_{x \to a^+} f(x) = L$ | $x$ approaches $a$ from values *greater than* $a$ |
| 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 don't match, two-sided limit DNE |
| Infinite limit at finite $a$ | $\lim_{x \to a} f(x) = \pm \infty$ | Describes unbounded behavior; limit does not exist as finite value; implies vertical asymptote at $x=a$ |
| Limit at infinity | $\lim_{x \to \pm \infty} f(x) = L$ | Describes long-run behavior as $x$ grows; implies horizontal asymptote at $y=L$ |
| Formal epsilon-delta definition | $\forall \varepsilon >0, \exists \delta>0 : 0<\|x-a\|<\delta \implies \|f(x)-L\|<\varepsilon$ | Always includes $0<\|x-a\|$ to exclude $x=a$ itself |
| Epsilon-delta shortcut for linear $f(x)=mx+b$ | $\delta = \frac{\varepsilon}{\|m\|}$ | Works for any linear function to find $\delta$ for a given $\varepsilon$ |

## What's next

This topic is the foundational language for all of calculus. Immediately after mastering limit definitions and notation, you will move on to estimating limits from graphs and tables, then calculating limits using algebraic techniques. Without a solid understanding of what limits are and how to interpret their notation, you cannot correctly understand core tested concepts like the derivative as a limit of a difference quotient or the definite integral as a limit of a Riemann sum. This topic also feeds into limits of sequences and series later in the course, which make up a large portion of the BC exam.

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

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