# Limits and Continuity Overview

> AP Calculus BC · Foundational first unit covering limits and continuity for AP Calculus BC
> Source: https://www.owlsprep.com/study/ap-calculus-bc-u1-overview/
> Weight: 10-12% of overall AP Calculus BC exam score

This unit introduces the foundational concept of limits, which underpins all of calculus, and connects limits to the core property of continuity. Mastery here sets the stage for derivatives and integrals later in the course.

**Prerequisites:** High school pre-calculus (proficiency with functions, graphs, and algebraic manipulation)

## Learning objectives

- Define limits using correct notation and interpret them from graphs, tables, and algebraic expressions
- Calculate limits using a range of algebraic, graphical, and theorem-based techniques
- Determine continuity at points and over intervals, and classify all types of discontinuities
- Apply the Intermediate Value Theorem to solve problems involving continuous functions

## 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?](https://www.owlsprep.com/study/ap-calculus-bc-u1-can-change-occur-at-an/) — Explores the core motivating question of calculus and introduces the idea of instantaneous change.
- [AP Calculus BC Confirming continuity over an interval](https://www.owlsprep.com/study/ap-calculus-bc-u1-confirming-continuity-over-an-interval/) — Learn to verify whether a function is continuous across an entire interval using limit properties.
- [AP Calculus BC Connecting infinite limits and vertical asymptotes](https://www.owlsprep.com/study/ap-calculus-bc-u1-connecting-infinite-limits-and-vertical/) — Connect infinite limits to the graphical behavior of vertical asymptotes.
- [AP Calculus BC Connecting limits at infinity and horizontal asymptotes](https://www.owlsprep.com/study/ap-calculus-bc-u1-connecting-limits-at-infinity-and/) — Link limits at infinity to the behavior of horizontal asymptotes for rational and other functions.
- [AP Calculus BC Connecting multiple representations of limits](https://www.owlsprep.com/study/ap-calculus-bc-u1-connecting-multiple-representations-of-limits/) — Practice translating between graphical, tabular, and algebraic representations of limits.
- [AP Calculus BC Defining continuity at a point](https://www.owlsprep.com/study/ap-calculus-bc-u1-defining-continuity-at-a-point/) — Formal definition of continuity at a single point in terms of limits.
- [AP Calculus BC Defining limits and using limit notation](https://www.owlsprep.com/study/ap-calculus-bc-u1-defining-limits-and-using-limit/) — Introduction to formal limit definition and correct limit notation.
- [AP Calculus BC Determining limits using algebraic manipulation](https://www.owlsprep.com/study/ap-calculus-bc-u1-determining-limits-using-algebraic-manipulation/) — Compute limits via factoring, conjugates, and other algebraic simplification techniques.
- [AP Calculus BC Determining limits using algebraic properties of limits](https://www.owlsprep.com/study/ap-calculus-bc-u1-determining-limits-using-algebraic-properties/) — Use properties of limits (sum, product, quotient, power) to compute basic limits.
- [AP Calculus BC Estimating limit values from graphs](https://www.owlsprep.com/study/ap-calculus-bc-u1-estimating-limit-values-from-graphs/) — Practice estimating limit values by reading and analyzing function graphs.
- [AP Calculus BC Estimating limit values from tables](https://www.owlsprep.com/study/ap-calculus-bc-u1-estimating-limit-values-from-tables/) — Estimate limit values from input-output tables of function values.
- [AP Calculus BC Exploring types of discontinuities](https://www.owlsprep.com/study/ap-calculus-bc-u1-exploring-types-of-discontinuities/) — Classify discontinuities as removable, jump, or infinite and describe their properties.
- [AP Calculus BC Removing discontinuities](https://www.owlsprep.com/study/ap-calculus-bc-u1-removing-discontinuities/) — Learn how to redefine functions to remove removable discontinuities.
- [AP Calculus BC Selecting procedures for determining limits](https://www.owlsprep.com/study/ap-calculus-bc-u1-selecting-procedures-for-determining-limits/) — Practice choosing the correct method to compute different types of limits.
- [AP Calculus BC Squeeze theorem](https://www.owlsprep.com/study/ap-calculus-bc-u1-squeeze-theorem/) — Learn how to apply the Squeeze Theorem to evaluate limits of bounded functions.
- [AP Calculus BC Working with the Intermediate Value Theorem (IVT)](https://www.owlsprep.com/study/ap-calculus-bc-u1-working-with-the-intermediate-value/) — Understand and apply the Intermediate Value Theorem (IVT) 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: The limit describes the behavior of $f(x)$ near $a$, not at $a$, so the two values can be entirely different
  - Correct: Always evaluate the limit separately from the function's value at the point of interest
- **Wrong:** Assuming all discontinuities can be removed with a point redefinition
  - Why it fails: Only point (removable) discontinuities can be removed; jump and infinite discontinuities cannot
  - Correct: Classify the type of discontinuity first before attempting to modify the function
- **Wrong:** Applying the Intermediate Value Theorem without confirming continuity on the interval
  - Why it fails: The IVT only holds for continuous functions on closed intervals, so it cannot be used for discontinuous functions
  - Correct: Always verify continuity on $[a,b]$ as a first step before applying the IVT

## Cheatsheet

| Concept / Formula | Key Notes |
| --- | --- |
| Algebraic Limit Properties | $\lim_{x \to a} [f(x) \pm g(x)] = \lim f(x) \pm \lim g(x)$, holds if both limits exist |
| $0/0$ Indeterminate Form | Use factoring or conjugate multiplication to remove the indeterminacy |
| Continuity at $x=a$ | Three requirements: $f(a)$ is defined, $\lim_{x \to a} f(x)$ exists, and the two values are equal |
| Infinite Limits | $\lim_{x \to a} f(x) = \pm \infty$ implies a vertical asymptote at $x=a$ |
| Limits at Infinity | $\lim_{x \to \pm \infty} f(x) = L$ implies a horizontal asymptote at $y=L$ |
| Squeeze Theorem | If $g(x) \leq f(x) \leq h(x)$ near $a$ and $\lim g = \lim h = L$, then $\lim f = L$ |
| Intermediate Value Theorem | If $f$ is continuous on $[a,b]$, $f$ takes every value between $f(a)$ and $f(b)$ 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.

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

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