AP Calculus AB Unit 1: Limits and Continuity

1.1 Can change occur at a single instant?  (Mega Smart Notes)

In this AP Calculus lesson, students explore the idea of instantaneous rate of change and how it can be interpreted both numerically and geometrically. These concepts provide the foundation for understanding limits, derivatives, and the behavior of functions in AP Calculus AB. 

▶ Video 1: Can Change occur at a single instant?
▶ Video 2: Geometric Interpretation of Instantaneous Rate of Change
▶ Video 3: Instantaneous Rate of Change – Worked Example 

1.2 Defining Limits and Using Limit Notation

In this AP Calculus lesson, students develop an intuitive understanding of limits and learn how to use limit notation. They explore one-sided limits, determine when a limit exists, and interpret limits from graphs and tables, building a strong foundation for the study of continuity and derivatives.

▶ Video 1: What Is a Limit?
▶ Video 2: One-Sided Limits & When Does a Limit Exist?
▶ Video 3: Reading Limits from Graphs
▶ Video 4: Reading Limits from Tables & Limit Notation

1.3 Estimating Limit Values from Graphs

In this AP Calculus lesson, students learn how to estimate limit values directly from graphs by interpreting open circles, closed points, asymptotes, and one-sided behavior. They also apply limit laws to products, shifted arguments, and composite functions, developing the graphical reasoning and problem-solving skills required for AP Calculus AB and BC.

▶ Video 1: Reading Limit Graphs — Open Circles, Closed Dots & Asymptotes
▶ Video 2: Does the Limit Exist? The Left Vs Right Test
▶ Video 3: Two graphs - One Limit! Products of Limits from graphs
▶ Video 4: Shifted and Composite Limits  

1.4 Estimating Limit Values from Tables

In this AP Calculus lesson, students learn how to estimate limit values from tables by analyzing function behavior from the left and right of a target value. The lesson explores convergence, jump discontinuities, unbounded and oscillating behavior, the difference between a limit and the actual function value, composite function limits, and situations where a table does not provide enough information to determine a limit.

▶ Video 1: 5 Limit Table patterns you should recognize 
▶ Video 2: When a Limit Table is Inconclusive?
▶ Video 3: Composite Limits from Tables

1.5 Determining Limits Using Algebraic Properties of Limits

In this AP Calculus lesson, students learn how to determine limits algebraically using the fundamental Limit Laws, including sum, difference, product, quotient, constant multiple, power, and root properties. Students also apply these properties to expressions involving multiple functions and analyze limits of piecewise functions by comparing one-sided limits, while distinguishing between a function's value at a point and its limiting behavior.

▶ Video 1: Limit Laws
▶ Video 2: Function Values Vs Limits
▶ Video 3: Piecewise Limits

1.6 Determining Limits Using Algebraic Manipulation

In this AP Calculus lesson, students learn how to determine limits using algebraic manipulation when direct substitution produces an indeterminate form. Techniques include factoring and canceling common factors, rationalizing radical expressions with conjugates, factoring powers of x, evaluating limits at infinity using dominant terms, analyzing absolute value and one-sided limits, applying fundamental trigonometric limits and identities, and solving limit problems involving unknown parameters.

▶ Video 1: Got indeterminate form 0/0? Factoring Limits
▶ Video 2: Got indeterminate form 0/0 with a root? Use Conjugate
▶ Video 3: Got indeterminate form 0/0 near x=0? Factor out the Lowest Power
▶ Video 4: Limits at infinity. Highest degree wins
▶ Video 5: Absolute Value Limits
▶ Video 6: Key Trigonometric Limit sinx over x
▶ Video 7: Finding Parameter from a Limit

1.7 Selecting Procedures for Determining Limits (Mega Smart Notes)

In this AP Calculus lesson, students learn how to select the appropriate procedure for determining limits based on the structure of the problem. Topics include one-sided limits with absolute value and piecewise functions, algebraic simplification of indeterminate forms, conjugates and factoring, trigonometric identities and fundamental trigonometric limits, and applying Limit Laws to determine unknown limits from given information.

▶ Video 1: Absolute Value Limits. Why you must check both sides
▶ Video 2: Got 0 over 0 when evaluating a Limit. Don't stop there
▶ Video 3: Got 0 over 0 in a Trig Limit? Use the right Identity
▶ Video 4: Can You Find the Missing Limit? Don't Fall for This Trap! 

1.8 Selecting Procedures for Determining Limits (Mega Smart Notes)

In this AP Calculus lesson, students learn how to determine limits using the Squeeze Theorem. They explore how a function can be trapped between two bounding functions that approach the same value, how to select appropriate inequalities, and how the theorem can determine limits involving bounded or oscillating functions. Worked examples and graphical representations help students recognize when the Squeeze Theorem applies and when it cannot be used.

▶ Video 1: The Squeeze Theorem explained
▶ Video 2: Which inequality actually works? 
▶ Video 3: Oscillating function with existing limit  

1.9 Connecting Multiple Representations of Limits (Mega Smart Notes)

In this AP Calculus lesson, students learn how to determine limits using the Squeeze Theorem. They explore how a function can be trapped between two bounding functions that approach the same value, how to select appropriate inequalities, and how the theorem can determine limits involving bounded or oscillating functions. Worked examples and graphical representations help students recognize when the Squeeze Theorem applies and when it cannot be used.

▶ Video 1: One Limit… 4 Completely Different Ways to See It!
▶ Video 2: Tables Vs Graphs
▶ Video 3: Oscillating function with existing limit