Lesson 2: 1-2 Defining Limits
Duration of Days: 1
Lesson Objective
The student will be able to:
Express the concept of a limit using correct mathematical notation.
Determine the limit of a function using graphical evidence and tables of values.
Identify that a limit describes the behavior of a function near a point, not necessarily at the point.
What does it mean for a function to "approach" a value?
Can a limit exist at x = c even if the function is undefined at x = c?
Why is it insufficient to look at only one side of a point when determining a general limit?
Limit
Limit Notation
One-Sided Limit (Left-hand/Right-hand) Approach
Existence of a Limit
Indeterminate Form
Function Value vs. Limit Value
MA.F-IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
Description: This lesson teaches students how to "read" the intent of a graph. They learn to ignore the specific point and instead look at where the "roads" (the graph) are leading from both the left and the right. If the roads meet at the same height L, the limit is L. If they don't meet, the limit does not exist.
Purpose: This lesson establishes the foundation for Continuity (1.11) and the Derivative (2.1). It is crucial for students to realize early on that f(c) and lim of c are two entirely different questions that happen to have the same answer only when a function is "well-behaved."
DOK Level 1 (Recall): Evaluating a limit by looking at a simple, continuous graph.
DOK Level 2 (Skill/Concept): Using a table of values (numerical approach) to estimate a limit.
DOK Level 3 (Strategic Thinking): Explaining why a limit does not exist using the "Three Reasons for Non-Existence" (Jump, Oscillation, Vertical Asymptote).
For Struggling Learners (Scaffolding):The Finger Trace: Have students put one finger on the graph to the left of c and one to the right, then slide them toward x=c. If their fingers "touch" at the same height, the limit exists.
Notation Template: Provide a fill-in-the-blank sentence: "As x gets closer to ____, the y-value gets closer to ____."
For Advanced Learners (Extension):Oscillating Behavior: Introduce the function to show a limit that fails to exist due to oscillation.
Piecewise Construction: Ask students to create their own piecewise function where the limit does not equal f(c).
AP College Board Assessments