Defining limits and using limit notation — AP Calculus BC
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.
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
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.
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 $L$.
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.
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.
5. AP-Style Concept Check ★★☆☆☆ ⏱ 3 min
Common Pitfalls
Why: Students rush to answer and forget that two-sided limits require matching one-sided limits by definition.
Why: Students confuse notation for describing unbounded behavior with the definition of an existing finite limit.
Why: Students confuse function evaluation with limit behavior, especially when working with continuous functions.
Why: Students forget that the limit does not depend on the value of $f$ at $x=a$, only on values near $a$.
Why: Both involve infinity in the notation, leading to mix-ups about the behavior being described.
Why: Students forget $|x - a|$ is a piecewise function that changes definition at $x=a$.