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: Direct Substitution gives 0 over 0. What should you do next?
▶ Video 4: A common trap students keep making

1.10 Exploring Types of Discontinuities (Mega Smart Notes)

In this lesson, students explore the different types of discontinuities and learn how to identify them from graphs and algebraic representations. They distinguish between removable, jump, and infinite discontinuities, connect each type to limit behavior, and determine when a discontinuity can be removed by redefining the function at a single point.

▶ Video 1: What Makes a Function Discontinuous?  3 Types You Need to Know
▶ Video 2: Removable Discontinuities
▶ Video 3: Jump Discontinuities Explained: When Left and Right Limits Disagree
▶ Video 4: Removable vs Infinite Discontinuities: Hole or Vertical Asymptote?
▶ Video 5: Discontinuities in Calculus: 3 Common Mistakes Students Make

1.11 Defining Continuity at a point (Mega Smart Notes)

Learn how to determine whether a function is continuous at a specific point using the three essential conditions for continuity. Explore continuity algebraically and graphically, work with piecewise functions, find missing parameters that make functions continuous, and apply continuity properties of standard functions and compositions.

▶ Video 1: Continuity at a Point: The 3 Conditions You MUST Check
▶ Video 2: Reading graphs for Continuity
▶ Video 3: Find missing parameters for continuity
▶ Video 4: Standard Continuous functions and compositions

1.12 Confirming Continuity over an Interval (Mega Smart Notes)

Learn how to determine whether a function is continuous at a specific point using the three essential conditions for continuity. Explore continuity algebraically and graphically, work with piecewise functions, find missing parameters that make functions continuous, and apply continuity properties of standard functions and compositions.

▶ Video 1: Continuity on Open vs Closed Intervals: The Endpoint Rule Explained
▶ Video 2: Which functions are continuous? Rules you need to know
▶ Video 3: Piecewise functions. How to find intervals of continuity
▶ Video 4: Absolute Value rational functions. Find the intervals of continuity
▶ Video 5: Continuity from a graph. How to find intervals of continuity

1.13 Removing Discontinuities (Mega Smart Notes)

Learn how to identify and remove removable discontinuities by using limits to determine the missing function value. Explore the difference between holes and non-removable discontinuities, use factoring to simplify rational functions, find unknown parameters that make functions continuous, and determine when no possible value can restore continuity.

▶ Video 1: Removable discontinuity. How to fix a hole
▶ Video 2: Removable vs Non-Removable Discontinuities: Factor First!
▶ Video 3: Find k for Continuity: Remove the Discontinuity Step by Step
▶ Video 4: 3 Piece Piecewise functions

1.14 Connecting Infinite Limits and Vertical Asymptotes (Mega Smart Notes)

In this Lesson you will learn how infinite limits are connected to vertical asymptotes and how to determine a function's behavior from each side of an asymptote. Explore rational and logarithmic functions, use sign analysis and one-sided limits, distinguish vertical asymptotes from holes, and interpret when a function approaches +∞ or −∞.

▶ Video 1: Infinite Limits & Vertical Asymptotes. The Connection Explained
▶ Video 2: Vertical asymptotes and infinite limits. How to find +infinity or -infinity
▶ Video 3: Hole Vs Vertical Asymptote. How to tell the difference
▶ Video 4: Vertical Asymptotes with Logarithms

1.15 Limits at Infinity and Horizontal Asymptotes (Mega Smart Notes)

Learn how to evaluate limits at infinity and use them to identify horizontal asymptotes and describe the long-term behavior of functions. Explore the degree-comparison rule for rational functions, compare the growth rates of logarithmic, polynomial, and exponential functions, and handle limits involving square roots and oscillating functions

▶ Video 1: Limits at Infinity & Horizontal Asymptotes: The Core Idea
▶ Video 2: Horizontal Asymptotes of Rational Functions: The Degree Rule
▶ Video 3: Limits at Infinity and Growth Hierarchy
▶ Video 4: Vertical Asymptotes with Logarithms