Unit Overview
Limits and Continuity Overview
AP Calculus BCΒ· 5 min read π 10-12% of overall AP Calculus BC exam score
1. Unit at a Glance
We open with the core question that motivates all of calculus: can change occur at an instant? This leads us to intuitively and formally define limits, then practice estimating limits from both graphs and tables.
After building intuition, we learn algebraic techniques to compute exact limits, then connect limits to the formal definition of continuity. We classify discontinuities, explore key theorems like the Intermediate Value Theorem, and finally connect limits to the asymptotic behavior of functions, tying together graphical and analytical representations.
This unit is organized into the following sub-topics:
AP Calculus BC Can change occur at an instant?
Explores the core motivating question of calculus and introduces the idea of instantaneous change.
β β± 3 min
AP Calculus BC Confirming continuity over an interval
Learn to verify whether a function is continuous across an entire interval using limit properties.
β β β± 3 min
AP Calculus BC Connecting infinite limits and vertical asymptotes
Connect infinite limits to the graphical behavior of vertical asymptotes.
β β β± 3 min
AP Calculus BC Connecting limits at infinity and horizontal asymptotes
Link limits at infinity to the behavior of horizontal asymptotes for rational and other functions.
β β β± 3 min
AP Calculus BC Connecting multiple representations of limits
Practice translating between graphical, tabular, and algebraic representations of limits.
β β β β± 4 min
AP Calculus BC Defining continuity at a point
Formal definition of continuity at a single point in terms of limits.
β β β± 3 min
AP Calculus BC Defining limits and using limit notation
Introduction to formal limit definition and correct limit notation.
β β± 3 min
AP Calculus BC Determining limits using algebraic manipulation
Compute limits via factoring, conjugates, and other algebraic simplification techniques.
β β β β± 4 min
AP Calculus BC Determining limits using algebraic properties of limits
Use properties of limits (sum, product, quotient, power) to compute basic limits.
β β β± 3 min
AP Calculus BC Estimating limit values from graphs
Practice estimating limit values by reading and analyzing function graphs.
β β± 2 min
AP Calculus BC Estimating limit values from tables
Estimate limit values from input-output tables of function values.
β β± 2 min
AP Calculus BC Exploring types of discontinuities
Classify discontinuities as removable, jump, or infinite and describe their properties.
β β β± 3 min
AP Calculus BC Removing discontinuities
Learn how to redefine functions to remove removable discontinuities.
β β β β± 3 min
AP Calculus BC Selecting procedures for determining limits
Practice choosing the correct method to compute different types of limits.
β β β β± 4 min
AP Calculus BC Squeeze theorem
Learn how to apply the Squeeze Theorem to evaluate limits of bounded functions.
β β β β β± 4 min
AP Calculus BC Working with the Intermediate Value Theorem (IVT)
Understand and apply the Intermediate Value Theorem (IVT) to continuous functions.
β β β β β± 4 min
2. Common Pitfalls
Wrong move:
Confusing the value of with the value of
Why:
The limit describes the behavior of near , not at , so the two values can be entirely different
Correct move:
Always evaluate the limit separately from the function's value at the point of interest
Wrong move:
Assuming all discontinuities can be removed with a point redefinition
Why:
Only point (removable) discontinuities can be removed; jump and infinite discontinuities cannot
Correct move:
Classify the type of discontinuity first before attempting to modify the function
Wrong move:
Applying the Intermediate Value Theorem without confirming continuity on the interval
Why:
The IVT only holds for continuous functions on closed intervals, so it cannot be used for discontinuous functions
Correct move:
Always verify continuity on as a first step before applying the IVT
3. Quick Reference Cheatsheet
Concept / Formula | Key Notes |
|---|---|
Algebraic Limit Properties | , holds if both limits exist |
Indeterminate Form | Use factoring or conjugate multiplication to remove the indeterminacy |
Continuity at | Three requirements: is defined, exists, and the two values are equal |
Infinite Limits | implies a vertical asymptote at |
Limits at Infinity | implies a horizontal asymptote at |
Squeeze Theorem | If near and , then |
Intermediate Value Theorem | If is continuous on , takes every value between and on the interval |
What's Next
Start with the first sub-topic of this unit to build your foundational understanding of limits, the backbone of all calculus. Work through each sub-topic in order to build up your skills step by step. Once you complete all topics in this unit, you will move on to differentiation, which directly builds on the limit and continuity concepts you master here.