Calculus AB · Unit 1: Limits and Continuity · 14 min read · Updated 2026-05-10
Defining limits and using limit notation — AP Calculus AB
AP Calculus AB · Unit 1: Limits and Continuity · 14 min read
1. Core Definition of a Limit and Two-Sided Notation★☆☆☆☆⏱ 4 min
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.
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.
2. One-Sided Limits and Existence Rule★★☆☆☆⏱ 4 min
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.
\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L
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.
3. Infinite Limits vs Limits at Infinity★★☆☆☆⏱ 3 min
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.
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.
4. AP-Style Concept Check★★☆☆☆⏱ 3 min
Common Pitfalls
Why: Students confuse which variable the direction applies to; direction is relative to $a$, not $x$.
Why: Students confuse the function's value at the point with the limit's value near the point.
Why: Students associate function value with limit value, so they assume no function value means no limit.
Why: Students confuse "infinity in the notation" with "infinite limit"; they think any limit with infinity does not exist.
Why: Students only check one one-sided limit instead of both, often matching the side of the function definition at $a$.
Why: Students see the equals sign and infinity written, so they incorrectly assume this means the limit exists as a finite value.