Lesson 4: 1-4 Finding Limits from Tables
Duration of Days: 1
Lesson Objective
The student will be able to:
Estimate the limit of a function by analyzing values in a table that approach the target x from both the left and the right.
Construct a table of values using a calculator to investigate limits that are difficult to see graphically.
Identify patterns in y-values (output) that suggest a limit exists, is infinite, or does not exist (DNE).
Determine the appropriate level of precision required to distinguish between a limit and a function's value.
If a table shows y-values of 2.9, 2.99, and 2.999, what value is the function likely approaching?
Why is it dangerous to look at only one or two points in a table to determine a limit?
How can a table reveal a "jump" or "oscillation" that might be hard to see on a standard graphing window?
Numerical Estimation
Delta
Convergence
Divergence
Monotonic Behavior
Arbitrarily Close
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 to "zoom in" numerically. They learn to evaluate a function at x-values, by observing the trend of the f(x) column, they can predict the limit L.
Purpose: Tables are a critical bridge between the "guessing" of graphs and the "certainty" of algebra. This method is particularly useful for "Indeterminate Forms" before students learn analytical techniques. It reinforces the idea that a limit is a trend, not a single point.
DOK Levels
DOK Level 1 (Recall): Identifying the limit from a pre-calculated table of values.
DOK Level 2 (Skill/Concept): Using a calculator to generate a table of values for a given function to estimate a limit.
DOK Level 3 (Strategic Thinking): Analyzing a table where the y-values are increasing without bound (identifying an infinite limit) or where the y-values do not converge to a single number.
For Struggling Learners (Scaffolding):The "Sandwich" Table: Provide a blank table template where the target x is in the center, forcing students to fill in values from both the left and right.
Decimal Alignment: Encourage students to write y-values vertically aligned to help them see which decimal place is "stabilizing.
"Calculus "Spy" Analogy: Tell students they are detectives looking for a secret meeting spot (the limit) by asking people standing nearby (the table values) where they are heading.
For Advanced Learners (Extension):
Hidden Oscillations: Provide a function and show how an poorly chosen x-increment in a table can lead to a false conclusion about the limit.
Rate of Convergence: Have students compare two different functions that both approach the same limit and determine which one "gets there" faster based on the table.
AP College Board Assessments