Skip to content

1.2 Defining Limits and Using Limit Notation

Keywords

Limit 极限 • Limit Notation 极限符号 • Real Number 实数

Arbitrarily Close 任意接近 • Sufficiently Close 充分接近

Graphically 图像上 • Numerically 数值上 • Analytically 解析/代数上

What is a Limit?

In Section 1.1, we learned that limits are the key to finding the instantaneous rate of change. But how do we formally define and write a limit?

Given a function \(f\), the limit (极限) of \(f(x)\) as \(x\) approaches \(c\) is a real number \(R\) if \(f(x)\) can be made arbitrarily close (任意接近) to \(R\) by taking \(x\) sufficiently close (充分接近) to \(c\).

Crucially, this happens when \(x\) gets close to \(c\), but not equal to \(c\) (\(x \neq c\)). A limit only cares about the journey (what the function is approaching), not the actual destination (the exact value of \(f(c)\)).

Limit Notation

If the limit exists and is a real number, then the common mathematical notation is:

\[ \lim_{x \to c} f(x) = R \]

This is read as: "The limit of \(f(x)\) as \(x\) approaches \(c\) equals \(R\)."

AP Exam Note / EXCLUSION STATEMENT The strict mathematical definition of a limit is known as the epsilon-delta (\(\epsilon-\delta\)) definition. This formal definition is not assessed on the AP Calculus AB or BC Exam. You only need to understand the concept of limits intuitively! (However, teachers may include this topic in the course if time permits).

Expressing Limits in Multiple Ways

A limit can be investigated and expressed in multiple ways. Throughout this course, you will learn to evaluate limits from three main perspectives:

  1. Numerically (数值上): Using a table of values to see what \(f(x)\) is approaching as \(x\) gets closer and closer to \(c\) from both sides (e.g., \(x=1.9, 1.99, 1.999...\)).
  2. Graphically (图像上): Looking at the graph of the function to visually determine the \(y\)-value the curve is heading towards as the \(x\)-value approaches \(c\).
  3. Analytically (解析上): Using algebra, theorems, and direct substitution to calculate the exact value of the limit.

(Note: In the upcoming sections, we will dive deep into each of these three methods!)

Quiz

Test your conceptual understanding of the definition of a limit.