Limits and Continuity

Students are introduced to calculus topics of limits, continuity, and derivatives, and gain a foundation of essential calculus skills.

Unit Summary

Unit 9, Limits and Continuity, is an introduction to the calculus topics of limits, continuity, and derivatives, and it provides a foundation in the essential calculus skill of thinking and reasoning about the infinitely small and the infinitely large while also arguing logically based on definitions and theorems. Students should leave this unit confident working with piecewise functions, finding finite and infinite limits of various types of functions graphically and algebraically, and rigorously defining continuity.

This unit also offers a useful opportunity to review and deepen knowledge of various types of functions, and rational functions in particular offer a useful opportunity to apply knowledge of limits, analyze asymptotes and removable discontinuities, and think about end behavior, while all function types help students to apply function transformations in different contexts. These skills are essential foundational knowledge for calculus, and this unit should be tailored to the specific strengths and weaknesses of students to maximize the utility of that flexibility. 

If there is not time for a nine-lesson unit at this point in the year, Lessons 3 and 5 could be cut. If there is extra time, Lesson 5 could be expanded to practice more parent functions, and Lessons 7 and 8 could be expanded to practice working with rational functions, factoring, and manipulating functions algebraically. If the unit is shortened, consider cutting corresponding questions from the assessment as well.

Since this unit serves as an introduction to Calculus, the lessons are aligned to the Learning Objectives and Essential Knowledge described in the College Board's AP Calculus AB and AP Calculus BC Course and Exam Description in the place of Common Core State Standards.

Pacing: 11 instructional days (9 lessons, 1 flex day, 1 assessment day)


This assessment accompanies Unit 9 and should be given on the suggested assessment day or after completing the unit.

Unit Prep

Intellectual Prep


Internalization of Standards via the Unit Assessment

  • Take unit assessment. Annotate for: 
    • Standards that each question aligns to
    • Purpose of each question: spiral, foundational, mastery, developing
    • Strategies and representations used in daily lessons
    • Relationship to Essential Questions of unit 
    • Lesson(s) that assessment points to

Internalization of Trajectory of Unit

  • Read and annotate “Unit Summary.”
  • Notice the progression of concepts through unit using the “Unit at a Glance.”
  • Do all target tasks. Annotate the target tasks for: 
    • Essential questions
    • Connection to assessment questions 
  • Answer the essential questions. (In the beginning, submit them to your instructional leader; toward the end, just bring them to the meeting.)

Essential Understandings


  • Limits allow us to analyze functions at places where they are not defined and to more rigorously describe the behavior of functions on infinitely small and infinitely large intervals.
  • Limits create an opportunity for a rigorous definition of continuity and more precisely defining a wider variety of functions, including piecewise functions.



Desmos: This unit requires generating numerous piecewise graphs with specific features, and Desmos is the easiest tool to create these graphs.



Limit Right-hand limit
Left-hand limit Continuous
Asymptote End behavior
Removable discontinuity Piecewise function
Parent function Transformation
Interval notation Slope graph
Open interval Closed interval
Limit at infinity  

Lesson Map

Topic A: Limits and Continuity


Graph, write, and evaluate linear piecewise functions.


Use interval and function notation to describe the behavior of piecewise functions.


Sketch a slope graph from a linear piecewise function.


Find limits, including left- and right-hand limits, on a function given graphically. 


Define continuity of functions and determine whether a function is continuous on a particular domain. 


Write and evaluate piecewise functions algebraically and graphically using parent functions.


State and evaluate limits algebraically.


Evaluate infinite limits and limits at infinity.


Sketch functions given limits and continuity requirements. 

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Standards for Mathematical Practice

  • CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.

  • CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.

  • CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.

  • CCSS.MATH.PRACTICE.MP4 — Model with mathematics.

  • CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.

  • CCSS.MATH.PRACTICE.MP6 — Attend to precision.

  • CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.

  • CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.