# Limits and Continuity Overview

> AP Calculus AB · Limits and Continuity
> Source: https://www.owlsprep.com/study/ap-calculus-ab-u1-overview/
> Weight: 10-12% of overall AP exam score

This unit introduces the core foundational concept of limits, which underlies all of calculus. You will connect limits to function continuity, build calculation skills, and prepare for differentiation and integration in later units.

**Prerequisites:** High school algebra, function notation, and graphing basics

## Learning objectives

- Define limits using formal notation and estimate limits from graphs and tables
- Calculate limits using algebraic properties, manipulation, and the Squeeze Theorem
- Classify discontinuities and determine continuity at points and over intervals
- Connect limits to function asymptotes and apply the Intermediate Value Theorem
- Build the foundational intuition required for all future calculus concepts

## 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?](https://www.owlsprep.com/study/ap-calculus-ab-u1-can-change-occur-at-an/) — Explore the core conceptual question that motivates the need for limits in calculus.
- [AP Calculus AB Defining limits and using limit notation](https://www.owlsprep.com/study/ap-calculus-ab-u1-defining-limits-and-using-limit/) — Master the formal definition of a limit and standard notation used throughout calculus.
- [AP Calculus AB Estimating limit values from graphs](https://www.owlsprep.com/study/ap-calculus-ab-u1-estimating-limit-values-from-graphs/) — Practice estimating limit values by analyzing the behavior of function graphs.
- [AP Calculus AB Estimating limit values from tables](https://www.owlsprep.com/study/ap-calculus-ab-u1-estimating-limit-values-from-tables/) — Estimate limit values by examining function values near the point of interest from tables.
- [AP Calculus AB Connecting multiple representations of limits](https://www.owlsprep.com/study/ap-calculus-ab-u1-connecting-multiple-representations-of-limits/) — Practice connecting limit information across graphs, tables, equations, and descriptions.
- [AP Calculus AB Determining limits using algebraic properties of limits](https://www.owlsprep.com/study/ap-calculus-ab-u1-determining-limits-using-algebraic-properties/) — Apply basic limit properties for sums, products, quotients, and compositions of functions.
- [AP Calculus AB Determining limits using algebraic manipulation](https://www.owlsprep.com/study/ap-calculus-ab-u1-determining-limits-using-algebraic-manipulation/) — Learn factoring and rationalizing to evaluate indeterminate $\frac{0}{0}$ limits.
- [AP Calculus AB Squeeze theorem](https://www.owlsprep.com/study/ap-calculus-ab-u1-squeeze-theorem/) — Apply the Squeeze Theorem to evaluate limits of bounded trigonometric functions.
- [AP Calculus AB Selecting procedures for determining limits](https://www.owlsprep.com/study/ap-calculus-ab-u1-selecting-procedures-for-determining-limits/) — Practice choosing the right evaluation strategy for any limit problem.
- [AP Calculus AB Connecting infinite limits and vertical asymptotes](https://www.owlsprep.com/study/ap-calculus-ab-u1-connecting-infinite-limits-and-vertical/) — Connect infinite limit behavior to the location of vertical asymptotes on function graphs.
- [AP Calculus AB Connecting limits at infinity and horizontal asymptotes](https://www.owlsprep.com/study/ap-calculus-ab-u1-connecting-limits-at-infinity-and/) — Relate end behavior of functions to limits at infinity and horizontal asymptotes.
- [AP Calculus AB Defining continuity at a point](https://www.owlsprep.com/study/ap-calculus-ab-u1-defining-continuity-at-a-point/) — Learn the formal three-part definition of continuity at a single point.
- [AP Calculus AB Exploring types of discontinuities](https://www.owlsprep.com/study/ap-calculus-ab-u1-exploring-types-of-discontinuities/) — Classify discontinuities as removable, jump, or infinite and identify them on graphs.
- [AP Calculus AB Removing discontinuities](https://www.owlsprep.com/study/ap-calculus-ab-u1-removing-discontinuities/) — Learn how to redefine functions to remove removable discontinuities algebraically.
- [AP Calculus AB Confirming continuity over an interval](https://www.owlsprep.com/study/ap-calculus-ab-u1-confirming-continuity-over-an-interval/) — Learn how to extend continuity checks from single points to entire intervals.
- [AP Calculus AB Working with the Intermediate Value Theorem (IVT)](https://www.owlsprep.com/study/ap-calculus-ab-u1-working-with-the-intermediate-value/) — Understand and apply the Intermediate Value Theorem to continuous functions.

## Common pitfalls

- **Wrong:** Confusing the value of $f(a)$ with the value of $
lim_{x \to a} f(x)$
  - Why it fails: $f(a)$ is the function's value at the point, while the limit describes behavior near the point, which can be very different.
  - Correct: Always separate the function value at $a$ from the limit near $a$ when solving problems.
- **Wrong:** Forgetting to check all three conditions for continuity at a point
  - Why it fails: Most students only check that the limit exists, missing that the limit must equal $f(a)$ and $f(a)$ must be defined.
  - Correct: Verify all three conditions explicitly: $f(a)$ is defined, $
lim_{x \to a} f(x)$ exists, and $
lim_{x \to a} f(x) = f(a)$.
- **Wrong:** Applying the Intermediate Value Theorem to discontinuous functions
  - Why it fails: IVT only holds for continuous functions on a closed interval, so misapplication leads to wrong conclusions.
  - Correct: Always confirm the function is continuous on the interval before citing the Intermediate Value Theorem.

## Cheatsheet

| Concept/Formula | Key Unit Summary |
| --- | --- |
| $ lim_{x \to a} f(x) = L$ | $f(x)$ approaches $L$ as $x$ gets arbitrarily close to (but not equal to) $a$ 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 $a$ | Three conditions: 1) $f(a)$ defined, 2) $ lim_{x \to a} f(x)$ exists, 3) $ lim_{x \to a} f(x) = f(a)$ |
| Removable discontinuity | Limit exists, but does not equal $f(a)$ (or $f(a)$ is undefined) |
| $ lim_{x \to a} f(x) =  pm  infty$ | Implies the graph has a vertical asymptote at $x=a$ |
| $ lim_{x \to  pm  infty} f(x) = L$ | Implies the graph has a horizontal asymptote at $y=L$ |
| Intermediate Value Theorem (IVT) | If $f$ is continuous on $[a,b]$, $f$ takes on every value between $f(a)$ and $f(b)$ |

## 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.

- [AP Calculus AB Can change occur at an instant?](https://www.owlsprep.com/study/ap-calculus-ab-u1-can-change-occur-at-an/)
- [Defining limits and using limit notation](https://www.owlsprep.com/study/ap-calculus-ab-u1-defining-limits-and-using-limit/)
- [Estimating limit values from graphs](https://www.owlsprep.com/study/ap-calculus-ab-u1-estimating-limit-values-from-graphs/)

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