Lesson 5: 1-5 Algebraic Properties of Limits and Piecewise Functions
Duration of Days: 1
Lesson Objective
The student will be able to:
Apply the basic properties of limits (sum, difference, product, quotient, and power) to evaluate limits of combined functions.
Evaluate the limit of a polynomial or rational function using direct substitution when the function is continuous
.Identify when direct substitution results in a defined value versus an indeterminate form.
Simplify expressions using algebraic properties to determine limits of functions as x approaches a constant.
If we know the limit of f(x) and the limit of g(x) separately, how can we find the limit of their sum or product?
When is it safe to simply "plug in" the value of c to find the limit?
Does the limit of a constant ever change, regardless of what x approaches?
Analytical Evaluation
Direct Substitution
Sum/Difference Property
Product/Quotient Property
Constant Multiple Property
Power/Root Property
Scalar
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 provides the algebraic "toolkit" for calculus. Instead of relying on a graph or a table, students learn that limits behave predictably. For example, the limit of a sum is the sum of the limits. Students learn that for most "well-behaved" functions (polynomials, sines, cosines), the limit is simply the function's value, which leads to the technique of Direct Substitution.
Purpose: This is the first step in solving limits with mathematical rigor. It bridges the gap between the intuitive "approaching" and the formal "calculating." It also sets the stage for the next few lessons, where direct substitution fails (resulting in 0/0), requiring more advanced algebraic manipulation.
DOK Levels
DOK Level 1 (Recall): Identifying which limit property to use for a given expression.
DOK Level 2 (Skill/Concept): Evaluating limits using direct substitution or by combining given limit values of f(x) and g(x).
DOK Level 3 (Strategic Thinking): Justifying each step of a limit evaluation using the formal names of the limit properties.
For Struggling Learners (Scaffolding):
Property "Matching" Game: Give students a complex limit and have them physically place cards with the names of the properties (e.g., "Product Rule") next to the corresponding steps.
For Advanced Learners (Extension):
Counter-Examples: Ask students to find a case where the Quotient Property for limits cannot be applied (i.e., when the limit of the denominator is zero).
AP College Board Assessments