Section outline

  • Developing Understanding

    Limits highlight the important difference between simply plugging in a value to a function and considering the value the function approaches as x nears a certain point. This subtlety lets us formally define concepts like asymptotes and holes in graphs, and lays the groundwork for understanding continuity. When introducing limits, it can be helpful to use rational functions as examples rather than starting with a comprehensive review of precalculus. Limits are essential for later topics—differentiation, integration, and infinite series in Unit 10 (BC-level only). They underpin key definitions, theorems, and methods used to solve real-world problems involving change and to justify mathematical conclusions.

    Building Mathematical Practices

    •    2.B
    •    2.C
    •    3.C
    •    3.D
    Mathematical information can be organized in various ways: graphically, numerically, analytically, or verbally. Mathematicians must communicate effectively across all these formats and move smoothly between them. Learning about limits helps students grow in these skills, showing them how to translate information from a table, equation, or text into a graph, and back again. Guidance should focus on matching and converting between analytical and verbal representations. Using graphing calculators to explore these connections is strongly encouraged.

    Mathematicians also build arguments and justify conclusions using precise definitions, theorems, and tests. A frequent student mistake is neglecting to state all relevant information before concluding a theorem. During Unit 1, it’s important to give students clear instruction and time to practice connecting the dots: they should first confirm that all conditions or hypotheses are met before making a conclusion.

    Preparing for the AP Exam

    This course should be a full-year journey towards mastering the material, culminating in the AP Exam. It’s important to develop both specific content knowledge and overall understanding. After completing Unit 1, students should be able to evaluate or estimate limits using graphs, tables, equations, or verbal descriptions.
    To avoid losing points on the AP Exam, students should regularly practice correct notation, show their setups, and properly round answers when using calculators. Remember, two test sections do not allow calculators, while the other two may require them. From the start of the course, stress the importance of stating and checking the hypotheses for theorems. Students should always make sure a theorem’s conditions are met before applying it, establishing sound mathematical habits early on.

    • Unit 1 on Khan Academy URL

      Follow the link to study unit 1 on the Khan Academy website.

    • 1.1 Introducing Calculus: Can Change Occur at an Instant?
      ENDURING UNDERSTANDING
      CHA-1 
      Calculus allows us to generalize knowledge about motion to diverse problems
      involving change.
       
      LEARNING OBJECTIVE

      CHA-1.A 

      Interpret the rate of change at an instant in terms of average rates of change over intervals containing that instant.

    • 1.2 Defining Limits and Using Limit Notation

      ENDURING UNDERSTANDING

      LIM-1
      Reasoning with definitions, theorems, and properties can be used to justify claims about limits.

      LEARNING OBJECTIVES

      LIM-1.A
      Represent limits analytically using correct notation.

      LIM-1.B
      Interpret limits expressed in analytic notation.

    • 1.3 Estimating Limit Values from Graphs

      ENDURING UNDERSTANDING
      LIM-1
      Reasoning with definitions, theorems, and properties can be used to justify claims about limits.

      LEARNING OBJECTIVE
      LIM-1.C
      Estimate limits of functions.

    • 1.4 Estimating Limit Values from Tables

      ENDURING UNDERSTANDING
      LIM-1
      Reasoning with definitions, theorems, and properties can be used to justify claims about limits.

      LEARNING OBJECTIVE
      LIM-1.C
      Estimate limits of functions.

    • 1.5 Determining Limits Using Algebraic Properties of Limits

      ENDURING UNDERSTANDING
      LIM-1
      Reasoning with definitions, theorems, and properties can be used to justify claims about limits.

      LEARNING OBJECTIVE
      LIM-1.D
      Determine the limits of functions using limit theorems.

    • 1.6 Determining Limits Using Algebraic Manipulation

      ENDURING UNDERSTANDING
      LIM-1
      Reasoning with definitions, theorems, and properties can be used to justify claims about limits.

      LEARNING OBJECTIVE
      LIM-1.E
      Determine the limits of functions using equivalent expressions for the function or the squeeze theorem.

    • 1.7 Selecting Procedures for Determining Limits

      This topic is intended to focus on the skill of selecting an appropriate procedure for determining limits. Students should be given opportunities to practice when and how to apply all learning objectives relating to determining limits. 

    • Calculating Limits Using the Limit Laws(251030_1) Assignment

      Calculating Limits Using the Limit Laws

    • 1.8 Determining Limits Using the Squeeze Theorem

      ENDURING UNDERSTANDING
      LIM-1
      Reasoning with definitions, theorems, and properties can be used to justify claims about limits.

      LEARNING OBJECTIVE
      LIM-1.E
      Determine the limits of functions using equivalent expressions for the function or the squeeze theorem.

    • 1.9 Connecting Multiple Representations of Limits

      This topic is intended to focus on connecting representations. Students should be given opportunities to practice when and how to apply all learning objectives relating to limits and translating mathematical information from a single representation or across multiple representations.

    • 1.10 Exploring Types of Discontinuities

      ENDURING UNDERSTANDING
      LIM-2
      Reasoning with definitions, theorems, and properties can be used to justify claims about continuity.

      LEARNING OBJECTIVE
      LIM-2.A
      Justify conclusions about continuity at a point using the definition.

    • 1.11 Defining Continuity at a Point

      ENDURING UNDERSTANDING
      LIM-2
      Reasoning with definitions, theorems, and properties can be used to justify claims about continuity.

      LEARNING OBJECTIVE
      LIM-2.A
      Justify conclusions about continuity at a point using the definition.

    • 1.12 Confirming Continuity over an Interval

      ENDURING UNDERSTANDING
      LIM-2
      Reasoning with definitions, theorems, and properties can be used to justify claims about continuity.

      LEARNING OBJECTIVE
      LIM-2.B
      Determine intervals over which a function is continuous.

    • Continuity, Discontinuity and IVT Notes File PDF
    • 1.13 Removing Discontinuities

      ENDURING UNDERSTANDING
      LIM-2
      Reasoning with definitions, theorems, and properties can be used to justify claims about continuity.

      LEARNING OBJECTIVE
      LIM-2.C
      Determine values of x or solve for parameters that make discontinuous functions continuous, if possible.

    • 1.14 Connecting Infinite Limits and Vertical Asymptotes

      ENDURING UNDERSTANDING
      LIM-2
      Reasoning with definitions, theorems, and properties can be used to justify claims about continuity.

      LEARNING OBJECTIVE
      LIM-2.D
      Interpret the behavior of functions using limits involving infinity.

    • Limit Exercise 1 File PDF
    • 1.15 Connecting Limits at Infinity and Horizontal Asymptotes

      ENDURING UNDERSTANDING
      LIM-2
      Reasoning with definitions, theorems, and properties can be used to justify claims about continuity.

      LEARNING OBJECTIVE
      LIM-2.D
      Interpret the behavior of functions using limits involving infinity.

    • Limit Exercise 2 (Barron's) File PDF
    • 1.16 Working with the Intermediate Value Theorem (IVT)

      ENDURING UNDERSTANDING
      FUN-1
      Existence theorems allow us to draw conclusions about a function’s behavior on an interval without precisely locating that behavior.

      LEARNING OBJECTIVE
      FUN-1.A
      Explain the behavior of a function on an interval using the Intermediate Value Theorem.