Back to Limits and Continuity

Lesson PreviewLesson 2 of 16 in this unit

1.2 Defining Limits and Using Limit Notation

Part of Limits and Continuity in AP® Calculus BC. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

Start free
A limit describes the value that a function approaches as the input approaches a certain number. The main idea is that calculus studies what happens near a point, not just exactly at the point. When we write
lim⁡x→af(x)=L\lim_{x \to a} f(x) = L
,
we mean that as x gets closer and closer to a, the outputs of f(x) get closer and closer to L. This does not require that f(a) actually equals L, and it does not even require that f(a) exists.

Limit notation gives us a precise way to describe behavior. A two-sided limit exists only if the function approaches the same value from both the left and the right. If the left-hand and right-hand behaviors are different, then the overall limit does not exist. This idea becomes the basis for continuity, derivatives, and many of the most important ideas in calculus.

Example 1: Find
lim⁡x→2(x2+3x−1)\lim_{x \to 2}(x^2+3x-1)
.
Since this is a polynomial, direct substitution works:
lim⁡x→2(x2+3x−1)=22+3(2)−1=4+6−1=9\lim_{x \to 2}(x^2+3x-1)=2^2+3(2)-1=4+6-1=9

Example 2: Suppose
f(x)={x+2,x≠310,x=3f(x)=\begin{cases} x+2, & x\ne3 \\ 10, & x=3 \end{cases}
Then the limit is based on nearby values, so
lim⁡x→3f(x)=lim⁡x→3(x+2)=5\lim_{x \to 3} f(x)=\lim_{x \to 3}(x+2)=5
even though f(3)=10

In this unit

16 lessons

  1. 1.1Introducing Calculus: Can Change Occur at an Instant?Preview lesson
  2. 1.2Defining Limits and Using Limit NotationThis lesson
  3. 1.3Estimating Limit Values from GraphsPreview lesson
  4. 1.4Estimating Limit Values from TablesPreview lesson
  5. 1.5Determining Limits Using Algebraic Properties of LimitsPreview lesson
  6. 1.6Determining Limits Using Algebraic ManipulationPreview lesson
  7. 1.7Selecting Procedures for Determining LimitsPreview lesson
  8. 1.8Determining Limits Using the Squeeze TheoremPreview lesson
  9. 1.9Connecting Multiple Representations of LimitsPreview lesson
  10. 1.10Exploring Types of DiscontinuitiesPreview lesson
  11. 1.11Defining Continuity at a PointPreview lesson
  12. 1.12Confirming Continuity over an IntervalPreview lesson
  13. 1.13Removing DiscontinuitiesPreview lesson
  14. 1.14Connecting Infinite Limits and Vertical AsymptotesPreview lesson
  15. 1.15Connecting Limits at Infinity and Horizontal AsymptotesPreview lesson
  16. 1.16Working with the Intermediate Value Theorem (IVT)Preview lesson

Continue the full lesson in Stellar

Create a free account to unlock the remaining sections, checkpoints, course practice, and progress tracking for AP® Calculus BC.

Start free