Lesson 14: 1-14 Infinite Limits and Vertical Asymptotes
Duration of Days: 2
Lesson Objective
The student will be able to:
Identify vertical asymptotes by finding x-values where the limit of a function is infinite.
Determine the direction of infinite limits using one-sided limit analysis.
Apply the "non-zero constant over zero" rule to verify the existence of a vertical asymptote.
Describe the behavior of functions near vertical asymptotes using proper limit notation.
Why is it more accurate to say a limit "is infinity" than to simply say it "Does Not Exist"?
How can a function have a vertical asymptote at x = c if the denominator is zero, but also have a hole there?
Infinite Limit
Vertical Asymptote
Unbounded Growth
One-Sided Infinite Limit
Rational Function Behavior
Convergence vs. Divergence
F-IF.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
Description: This lesson formalizes the "infinite" nature of some limits. Students learn that a vertical asymptote exists at x = c if and only if the limit from either the left or the right. Unlike a hole (removable), this behavior is an essential part of the graph that cannot be "fixed." Students use sign analysis ( to determine if the function shoots up to infinity or down to infinity.
Purpose: Infinite limits are the analytical foundation for understanding the shape of rational, logarithmic, and certain trigonometric functions. Mastering this allows students to accurately sketch graphs and understand the physical constraints of mathematical models.
DOK Level 2 (Skill/Concept): Identifying all vertical asymptotes of a rational function and determining the behavior from both sides.
DOK Level 3 (Strategic Thinking): Explaining the difference in behavior between functions near the asymptote x=2.
For Struggling Learners (Scaffolding):
Sign Analysis Table: Encourage a simple +/- check for the numerator and denominator to determine the final sign of infinity.
For Advanced Learners (Extension):Limits of Transcendental Functions: Have students analyze limits.
AP College Board Assessments