Defining limits and using limit notation

AP Calculus BC· AP Calculus BC CED — Limits and Continuity· 14 min read

1. Core Definition of Limits and Standard Notation★☆☆☆☆⏱ 3 min

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.

📘 Definition

Limit

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

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 $a$ .

2. One-Sided and Two-Sided Limits★★☆☆☆⏱ 4 min

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 $f (x) = {2 x + 1 x^{2} - 2 x x < 3 x > 3$ , find $lim_{x \to 3} f (x)$ if it exists.

1
Calculate the left-hand limit as $x$ approaches 3 from the left. For $x < 3$ , $f (x) = 2 x + 1$ :
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 $x > 3$ , $f (x) = x^{2} - 2 x$ :
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.

3. Infinite Limits and Vertical Asymptotes★★☆☆☆⏱ 3 min

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 $lim_{x \to 2^{-}} \frac{3 x + 1}{x - 2}$ , 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.

4. Formal Epsilon-Delta Definition of a Limit★★★☆☆⏱ 4 min

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 $L$ .

📘 Definition

Epsilon-Delta Limit Definition

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

For every $ε > 0$ , there exists a $δ > 0$ such that if $0 < ∣ x - a ∣ < δ$ , then $∣ f (x) - L ∣ < ε$ . $ε$ is the maximum allowed error between $f (x)$ and $L$ , and $δ$ is how close $x$ must be to $a$ (excluding $x = a$ itself) to satisfy the error bound.

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 $δ$ for a given $ε$ on multiple-choice questions.

📐 Worked Example

For $lim_{x \to 4} (3 x - 5) = 7$ , find $δ$ that satisfies the epsilon-delta condition for $ε = 0.1$ .

1
The definition requires that $∣ (3 x - 5) - 7∣ < 0.1$ whenever $0 < ∣ x - 4∣ < δ$ .
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 $δ = 0.0333$ (or any smaller positive number) satisfies the condition for $ε = 0.1$ .

Exam tip:

When finding $δ$ for a linear function $f (x) = m x + b$ , $δ$ will always equal $\frac{ε}{∣ m ∣}$ —use this shortcut to save time on MCQ.

5. AP-Style Concept Check★★☆☆☆⏱ 3 min

✓ Quick check

Test your understanding of limit definitions and notation:

Given $g (x) = \frac{∣ x - 4∣}{x - 4}$ , what is $lim_{x \to 4} g (x)$ ?
- 0
- 1
- -1
- The limit does not exist
Reveal answer
The limit does not exist —
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.
Let $h (x) = {x^{2} + k 3 x + 2 k x \leq 2 x > 2$ . For what value of $k$ does $lim_{x \to 2} h (x)$ exist?
- k=-2
- k=2
- k=-1
- k=1
Reveal answer
k=-2 —
Correct. Set the one-sided limits equal: $4 + k = 6 + 2 k ⟹ k = - 2$ .
An ecologist models deer population as $P (t) = \frac{1200 t}{t + 5}$ for $t \geq 0$ , where carrying capacity is 1200. What is the correct limit notation for long-term population behavior?
- $lim_{t \to 1200} P (t) = \infty$
- $lim_{t \to \infty} P (t) = 1200$
- $lim_{P \to 1200} t = \infty$
- $lim_{t \to 1200} P (t) = t$
Reveal answer
$\lim_{t \to \infty} P(t) = 1200$ —
Correct. Long-term behavior means $t$ (time) approaches infinity, and the limit approaches the carrying capacity of 1200.

6. Common Pitfalls

Wrong move:

After calculating only one one-sided limit for a piecewise function at the boundary, concluding the two-sided limit equals that value.

Why:

Students rush to answer and forget that two-sided limits require matching one-sided limits by definition.

Correct move:

Always explicitly calculate left-hand and right-hand limits, then compare them before concluding whether the two-sided limit exists.

Wrong move:

Stating that $lim_{x \to a} f (x) = \infty$ means the limit exists.

Why:

Students confuse notation for describing unbounded behavior with the definition of an existing finite limit.

Correct move:

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 move:

Evaluating $f (a)$ to find $lim_{x \to a} f (x)$ , and concluding the limit equals $f (a)$ .

Why:

Students confuse function evaluation with limit behavior, especially when working with continuous functions.

Correct move:

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 move:

In epsilon-delta notation, dropping the $0 < ∣ x - a ∣$ condition and writing only $∣ x - a ∣ < δ$ .

Why:

Students forget that the limit does not depend on the value of $f$ at $x = a$ , only on values near $a$ .

Correct move:

Always include the $0 <$ inequality in epsilon-delta statements to exclude $x = a$ itself.

Wrong move:

Confusing $lim_{x \to \infty} f (x)$ (limit at infinity) with $lim_{x \to a} f (x) = \infty$ (infinite limit at finite $a$ ).

Why:

Both involve infinity in the notation, leading to mix-ups about the behavior being described.

Correct move:

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 move:

For absolute value functions, using the same expression for left and right limits at the critical point.

Why:

Students forget $∣ x - a ∣$ is a piecewise function that changes definition at $x = a$ .

Correct move:

Rewrite $∣ x - a ∣$ into its piecewise form before calculating one-sided limits at $x = a$ .

7. Quick Reference 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	$lim_{x \to a} f (x) = L ⟺ lim_{x \to a^{-}} f (x) = lim_{x \to a^{+}} f (x) = L$	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	$\forall ε > 0, \exists δ > 0 : 0 < ∣ x - a ∣ < δ ⟹ ∣ f (x) - L ∣ < ε$	Always includes $0 < ∣ x - a ∣$ to exclude $x = a$ itself
Epsilon-delta shortcut for linear $f (x) = m x + b$	$δ = \frac{ε}{∣ m ∣}$	Works for any linear function to find $δ$ for a given $ε$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2025 · MCQ
Two-sided limit for piecewise function
2024 · FRQ
Interpret limit in context

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.