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.