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Unit 1: Limits and Continuity

16 dot points across 16 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How can a quantity have a rate of change at a single instant, when change seems to require an interval of time?

Topic 1.1 Introducing Calculus - Can Change Occur at an Instant?: understand how the idea of an instantaneous rate of change motivates the limit, and how average rates of change over shrinking intervals approach it.
A focused answer to AP Calculus AB Topic 1.1, explaining how average rates of change over shrinking intervals motivate the instantaneous rate of change and the limit, with worked difference-quotient examples.
9 min answer →

How do the graphical, numerical, and algebraic views of a limit fit together into one consistent answer?

Topic 1.9 Connecting Multiple Representations of Limits: translate among graphical, numerical, analytical and verbal representations of a limit and confirm they agree.
A focused answer to AP Calculus AB Topic 1.9, showing how graphical, numerical, algebraic and verbal representations of a limit describe the same value, with a worked cross-check example.
8 min answer →

What exactly must be true for a function to be continuous at a single point?

Topic 1.11 Defining Continuity at a Point: state and apply the three-part definition of continuity at a point and test functions against it.
A focused answer to AP Calculus AB Topic 1.11, giving the three-part definition of continuity at a point and applying it to piecewise functions, including solving for a parameter that makes a function continuous.
8 min answer →

What does it mean for a function to be continuous over an entire interval, and which familiar functions qualify?

Topic 1.12 Confirming Continuity over an Interval: determine intervals on which a function is continuous, using one-sided continuity at endpoints and the continuity of standard function families.
A focused answer to AP Calculus AB Topic 1.12, defining continuity over open and closed intervals, the continuity of polynomial, rational, root, trig, exponential and log families, and one-sided continuity at endpoints.
8 min answer →

What does it mean for a function to approach a value, and how do we write that idea precisely?

Topic 1.2 Defining Limits and Using Limit Notation: express the limit of a function using correct notation, including one-sided limits, and interpret what a limit says about the behavior of a function near a point.
A focused answer to AP Calculus AB Topic 1.2, defining the limit of a function, two-sided versus one-sided limits, correct limit notation, and when a limit fails to exist, with worked examples.
9 min answer →

How do you read the limit of a function at a point directly from its graph, even where the function is undefined?

Topic 1.3 Estimating Limit Values from Graphs: use a graph to estimate one-sided and two-sided limits, including cases where the function value differs from the limit or does not exist.
A focused answer to AP Calculus AB Topic 1.3, showing how to read one-sided and two-sided limits from a graph, distinguish the limit from the function value, and recognize holes, jumps and asymptotes.
8 min answer →

How can a table of function values let you estimate a limit you cannot evaluate directly?

Topic 1.4 Estimating Limit Values from Tables: use a table of values approaching a point from both sides to estimate one-sided and two-sided limits.
A focused answer to AP Calculus AB Topic 1.4, showing how to estimate one-sided and two-sided limits from a table of values, including the indeterminate-form case, with a fully worked example.
8 min answer →

What does an infinite limit tell you about a function, and how does it locate a vertical asymptote?

Topic 1.14 Connecting Infinite Limits and Vertical Asymptotes: use infinite limits to identify vertical asymptotes and describe behavior near them with correct sign analysis.
A focused answer to AP Calculus AB Topic 1.14, connecting infinite one-sided limits to vertical asymptotes, with sign analysis to determine whether the function goes to plus or minus infinity, and worked examples.
8 min answer →

How does continuity guarantee that a function must hit every value between its endpoints?

Topic 1.16 Working with the Intermediate Value Theorem (IVT): state the hypotheses of the IVT and use it to guarantee the existence of a value or a root on a closed interval.
A focused answer to AP Calculus AB Topic 1.16, stating the Intermediate Value Theorem, its continuity hypothesis, and using it to guarantee a root or a target value on a closed interval, with a full worked justification.
8 min answer →

What happens to a function as the input grows without bound, and how does that reveal a horizontal asymptote?

Topic 1.15 Connecting Limits at Infinity and Horizontal Asymptotes: evaluate limits as x approaches plus or minus infinity and use them to identify horizontal asymptotes, especially for rational functions.
A focused answer to AP Calculus AB Topic 1.15, evaluating limits as x approaches infinity, the degree rule for rational functions, and identifying horizontal asymptotes, with worked examples.
9 min answer →

When direct substitution gives the indeterminate form 0/0, how do you rewrite the function to find the limit?

Topic 1.6 Determining Limits Using Algebraic Manipulation: resolve indeterminate forms by factoring and cancelling, rationalizing, combining fractions, or using known trigonometric limits.
A focused answer to AP Calculus AB Topic 1.6, covering how to resolve 0/0 indeterminate forms by factoring, rationalizing and combining fractions, plus the key trigonometric limits, with full worked examples.
9 min answer →

How do the limit laws let you break a complicated limit into simple pieces you can evaluate?

Topic 1.5 Determining Limits Using Algebraic Properties of Limits: apply the limit laws (sum, difference, product, quotient, constant multiple, power) and direct substitution to evaluate limits.
A focused answer to AP Calculus AB Topic 1.5, covering the limit laws (sum, product, quotient, power) and direct substitution for evaluating limits of continuous functions, with worked examples.
8 min answer →

When a function has a hole, how do you redefine it at that point to make it continuous?

Topic 1.13 Removing Discontinuities: recognize a removable discontinuity and define or redefine the function value to make it continuous.
A focused answer to AP Calculus AB Topic 1.13, showing how to remove a removable discontinuity by assigning the limit value, and why jump and infinite discontinuities cannot be removed, with worked examples.
8 min answer →

Given any limit, how do you decide which method - substitution, algebra, a table, or a graph - to use?

Topic 1.7 Selecting Procedures for Determining Limits: choose an efficient strategy for a given limit, recognizing which technique fits the form of the function.
A focused answer to AP Calculus AB Topic 1.7, a decision strategy for choosing the right limit technique (substitution, factoring, conjugates, special trig limits, tables or graphs) based on the form of the function.
8 min answer →

How can you find a limit by trapping a function between two others that share the same limit?

Topic 1.8 Determining Limits Using the Squeeze Theorem: apply the squeeze (sandwich) theorem to evaluate limits of functions bounded between two functions with a common limit.
A focused answer to AP Calculus AB Topic 1.8, stating the squeeze (sandwich) theorem and applying it to limits like x squared times sine of one over x, with a full worked example.
8 min answer →

What are the different ways a function can fail to be continuous, and how do you tell them apart?

Topic 1.10 Exploring Types of Discontinuities: classify discontinuities as removable (point/hole), jump, or infinite (asymptotic), using limits.
A focused answer to AP Calculus AB Topic 1.10, classifying removable (hole), jump and infinite (asymptotic) discontinuities using one-sided and two-sided limits, with worked identification.
8 min answer →