Unit Overview
Limits and Continuity Overview
AP Calculus ABΒ· 5 min read π 10-12% of overall AP exam score
1. Unit at a Glance
Limits are the idea that lets calculus answer questions about instantaneous change, which cannot be solved with algebra alone. This unit progresses from intuitive conceptual introductions, to formal limit calculation techniques, to connecting limits to continuity and key calculus theorems.
Mastery of this unit is non-negotiable: every future concept in AP Calculus AB (including derivatives and integrals) is formally defined using limits. The sub-topics below are ordered to build your knowledge incrementally from basics to application.
All sub-topics in this unit are listed below in learning order:
AP Calculus AB Can change occur at an instant?
Explore the core conceptual question that motivates the need for limits in calculus.
β β± 3 min
AP Calculus AB Defining limits and using limit notation
Master the formal definition of a limit and standard notation used throughout calculus.
β β β± 4 min
AP Calculus AB Estimating limit values from graphs
Practice estimating limit values by analyzing the behavior of function graphs.
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AP Calculus AB Estimating limit values from tables
Estimate limit values by examining function values near the point of interest from tables.
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AP Calculus AB Connecting multiple representations of limits
Practice connecting limit information across graphs, tables, equations, and descriptions.
β β β± 3 min
AP Calculus AB Determining limits using algebraic properties of limits
Apply basic limit properties for sums, products, quotients, and compositions of functions.
β β β± 4 min
AP Calculus AB Determining limits using algebraic manipulation
Learn factoring and rationalizing to evaluate indeterminate limits.
β β β β± 5 min
AP Calculus AB Squeeze theorem
Apply the Squeeze Theorem to evaluate limits of bounded trigonometric functions.
β β β β β± 5 min
AP Calculus AB Selecting procedures for determining limits
Practice choosing the right evaluation strategy for any limit problem.
β β β β± 4 min
AP Calculus AB Connecting infinite limits and vertical asymptotes
Connect infinite limit behavior to the location of vertical asymptotes on function graphs.
β β β± 4 min
AP Calculus AB Connecting limits at infinity and horizontal asymptotes
Relate end behavior of functions to limits at infinity and horizontal asymptotes.
β β β β± 4 min
AP Calculus AB Defining continuity at a point
Learn the formal three-part definition of continuity at a single point.
β β β± 4 min
AP Calculus AB Exploring types of discontinuities
Classify discontinuities as removable, jump, or infinite and identify them on graphs.
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AP Calculus AB Removing discontinuities
Learn how to redefine functions to remove removable discontinuities algebraically.
β β β β± 4 min
AP Calculus AB Confirming continuity over an interval
Learn how to extend continuity checks from single points to entire intervals.
β β β± 4 min
AP Calculus AB Working with the Intermediate Value Theorem (IVT)
Understand and apply the Intermediate Value Theorem to continuous functions.
β β β β β± 5 min
2. Common Pitfalls
Wrong move:
Confusing the value of with the value of $ lim_{x \to a} f(x)$
Why:
is the function's value at the point, while the limit describes behavior near the point, which can be very different.
Correct move:
Always separate the function value at from the limit near when solving problems.
Wrong move:
Forgetting to check all three conditions for continuity at a point
Why:
Most students only check that the limit exists, missing that the limit must equal and must be defined.
Correct move:
Verify all three conditions explicitly: is defined, $ lim_{x \to a} f(x) lim_{x \to a} f(x) = f(a)$.
Wrong move:
Applying the Intermediate Value Theorem to discontinuous functions
Why:
IVT only holds for continuous functions on a closed interval, so misapplication leads to wrong conclusions.
Correct move:
Always confirm the function is continuous on the interval before citing the Intermediate Value Theorem.
3. Quick Reference Cheatsheet
Concept/Formula | Key Unit Summary |
|---|---|
$ lim_{x \to a} f(x) = L$ | approaches as gets arbitrarily close to (but not equal to) from both sides |
Basic limit properties | $ lim (c f) = c lim f lim (f pm g) = lim f pm lim g lim (fg) = ( lim f)( lim g)$ |
Indeterminate $ frac{0}{0}$ | Use factoring, conjugates/rationalizing, or trig identities to simplify before evaluating |
Continuity at | Three conditions: 1) defined, 2) $ lim_{x \to a} f(x) lim_{x \to a} f(x) = f(a)$ |
Removable discontinuity | Limit exists, but does not equal (or is undefined) |
$ lim_{x \to a} f(x) = pm infty$ | Implies the graph has a vertical asymptote at |
$ lim_{x \to pm infty} f(x) = L$ | Implies the graph has a horizontal asymptote at |
Intermediate Value Theorem (IVT) | If is continuous on , takes on every value between and |
What's Next
Start your learning of this unit with the first sub-topic below, which introduces the core question that led to the invention of calculus. Work through each sub-topic in order to build your knowledge incrementally. After you complete all sub-topics in this unit, you will move on to differentiation, the next core unit of AP Calculus AB.