Lesson 15: 1-15 Limits at Infinity and Horizontal Asymptotes
Duration of Days: 2
Lesson Objective
The student will be able to:
Evaluate limits as x approaches infinity using algebraic techniques and hierarchy of growth.
Connect the numerical result of a limit at infinity to the existence of a horizontal asymptote.
Apply the "Degree Test" (comparing the highest powers) for rational functions.
Identify horizontal asymptotes for non-rational functions, such as exponential and logistic functions.
As x becomes extremely large, which terms in a polynomial "take over" and which ones become insignificant?
Can a function have more than one horizontal asymptote?
What is the difference between a horizontal asymptote and a vertical asymptote in terms of the function "touching" or "crossing" the line?
Limit at Infinity
Horizontal Asymptote
End Behavior
Relative Growth Rates
Dominant Term
Convergence to a Constant
F-IF.7: Graph functions expressed symbolically and show key features of the graph.
Description: This lesson teaches students to look at the "big picture.".
Purpose: Understanding end behavior is crucial for modeling real-world phenomena, such as terminal velocity in physics or carrying capacity in biology. It also completes the student's ability to describe the global behavior of any function.
DOK Level 2 (Skill/Concept): Finding the horizontal asymptote of a rational function using the degree of the numerator and denominator.
DOK Level 3 (Strategic Thinking): Determining limits at infinity for functions involving square roots, which often have different asymptotes.
For Struggling Learners (Scaffolding):The "Million Dollar" Analogy: If you have a billion dollars and someone gives you five dollars , the five dollars doesn't really matter. Only the highest power (the "boss") determines the limit.
For Advanced Learners (Extension):
Briefly introduce slant (oblique) asymptotes for top-heavy functions where the degree is exactly one higher in the numerator.
AP College Board AP Assessments