# Removing Discontinuities

> AP Calculus BC · Unit 1: Limits and Continuity
> Source: https://www.owlsprep.com/study/ap-calculus-bc-u1-removing-discontinuities/

This module covers identification of removable discontinuities (holes), algebraic evaluation of limits at these points, and redefining functions to remove discontinuities for AP Calculus BC Unit 1, aligned to College Board exam expectations.

**Prerequisites:** [Evaluating one-sided and two-sided limits algebraically and graphically](https://www.owlsprep.com/study/ap-calculus-bc-u1-evaluating-limits-algebraically/); Factoring polynomials and recognizing special products; [Formal definition of continuity at a point](https://www.owlsprep.com/study/ap-calculus-bc-u1-continuity-at-a-point/)

## Learning objectives

- Identify removable vs non-removable discontinuities
- Evaluate limits at removable discontinuities using algebraic methods
- Redefine functions to remove removable discontinuities
- Solve common AP-style exam questions on this topic

## What Is a Removable Discontinuity?

A removable discontinuity is a point $x=a$ where a function $f(x)$ is discontinuous, but the two-sided limit $\lim_{x \to a} f(x)$ exists and is finite. The discontinuity arises only because $f(a)$ is undefined, or $f(a)$ is defined but not equal to this limit. 'Removing' the discontinuity means redefining $f(a)$ to equal the existing limit, resulting in a function that is continuous at $x=a$.

According to the AP Calculus CED, this topic falls within Unit 1: Limits and Continuity, which accounts for 10-12% of the total AP exam score. Removable discontinuities are most commonly tested in multiple-choice questions, but can also appear as a component of free-response questions testing continuity or piecewise function definitions.

**Removable Discontinuity** — A discontinuity at $x=a$ where the two-sided limit $\lim_{x \to a} f(x)$ exists and is finite. The discontinuity can be removed by redefining $f(a)$ to equal this limit.

*Notation:* Also called a 'hole' or 'point discontinuity'

*Example:* A rational function with a common factor in numerator and denominator has a removable discontinuity at the root of that common factor.

The key property that makes a discontinuity removable is the existence of a finite two-sided limit, which differentiates it from non-removable types (jump, infinite, oscillating) where the two-sided limit does not exist.

*Calculator:* allowed

## Identifying Removable Discontinuities

The formal test for a removable discontinuity at $x=a$ has two requirements: 1) $f(x)$ is discontinuous at $x=a$, and 2) $\lim_{x \to a} f(x)$ exists and is finite. For rational functions (the most common context on the AP exam), removable discontinuities occur when there is a common shared factor between the numerator and denominator. The root of this common factor is the location of the removable discontinuity.

To identify removable discontinuities for a rational function $f(x) = \frac{N(x)}{D(x)}$, follow two core steps: first, find all points where $D(x)=0$ (these are all points of discontinuity). Second, for each such point, check if the two-sided limit exists and is finite. If yes, it is removable; if not, it is non-removable (usually an asymptote). It is critical to check every point of discontinuity, not just assume all are removable.

**Worked example:** Identify all removable discontinuities of $f(x) = \frac{x^3 - 2x^2 - 8x}{x^2 - 6x + 8}$.

1. 1. Factor the numerator and denominator completely:
2. $$x^3 - 2x^2 - 8x = x(x^2 - 2x - 8) = x(x-4)(x+2) \\ x^2 - 6x + 8 = (x-2)(x-4)$$
3. 2. Find all points of discontinuity: The denominator equals zero at $x=2$ and $x=4$, so $f(x)$ is undefined (discontinuous) at both points.
4. 3. Evaluate the limit at each point:
5. For $x \to 2$: After canceling common factors, $\lim_{x \to 2} \frac{x(x+2)}{x-2} = \frac{2(4)}{0}$, so the limit is infinite and does not exist.
6. For $x \to 4$: After canceling $(x-4)$, $\lim_{x \to 4} \frac{x(x+2)}{x-2} = \frac{4(6)}{2} = 12$, a finite two-sided limit.
7. 4. Conclusion: Only $x=4$ is a removable discontinuity.

> **tip**
>
> On AP MCQ questions asking to count removable discontinuities, always check every root of the denominator, even if one is obviously removable. It is common for questions to include one removable and one non-removable discontinuity to test if you check all points.

*Calculator:* allowed

## Evaluating Limits at Removable Discontinuities

The most common AP exam skill tested on this topic is evaluating the limit at a removable discontinuity, a core Unit 1 skill. When you have a $0/0$ indeterminate form (the signature of a removable discontinuity), you use algebraic manipulation to eliminate the common factor causing the zero in the denominator, then evaluate the resulting limit by substitution.

The intuition behind this technique is that the limit as $x \to a$ only depends on the behavior of $f(x)$ near $x=a$, not at $x=a$. Canceling the common $(x-a)$ term does not change the value of $f(x)$ for any $x \neq a$, so the limit remains unchanged. Common algebraic techniques are factoring (for polynomials), multiplying by the conjugate (for functions with square roots), and simplifying complex fractions.

**Worked example:** Evaluate $\lim_{x \to 16} \frac{4 - \sqrt{x}}{16 - x}$, which has a removable discontinuity at $x=16$.

1. 1. Confirm the indeterminate form: At $x=16$, numerator = $4 - 4 = 0$, denominator = $16 - 16 = 0$, so we have a $0/0$ form indicating a potential removable discontinuity.
2. 2. Use the conjugate method to eliminate the square root. Multiply numerator and denominator by the conjugate of the numerator, $4 + \sqrt{x}$:
3. $$\lim_{x \to 16} \frac{(4 - \sqrt{x})(4 + \sqrt{x})}{(16 - x)(4 + \sqrt{x})}$$
4. 3. Simplify the numerator: $(4 - \sqrt{x})(4 + \sqrt{x}) = 16 - x$, so the expression becomes:
5. $$\lim_{x \to 16} \frac{16 - x}{(16 - x)(4 + \sqrt{x})}$$
6. 4. Cancel the common $(16-x)$ term (valid because $x \to 16$ means $x \neq 16$, so $16-x \neq 0$) to get:
7. $$\lim_{x \to 16} \frac{1}{4 + \sqrt{x}}$$
8. 5. Evaluate by substitution: $\frac{1}{4 + \sqrt{16}} = \frac{1}{8}$. The limit is $\frac{1}{8}$.

> **tip**
>
> When you have a 0/0 indeterminate form, always try algebraic manipulation first. Even if you know L'Hospital's Rule, AP Unit 1 questions explicitly test this algebraic technique, and you may be required to show algebraic work to earn full credit on FRQ.

*Calculator:* allowed

## Removing Discontinuities by Redefining the Function

Once you have confirmed that $x=a$ is a removable discontinuity and calculated $\lim_{x \to a}f(x) = L$, removing the discontinuity means redefining $f(a)$ to equal $L$, which makes the function continuous at $x=a$. This is the origin of the term 'removing' the discontinuity: we eliminate the discontinuity by fixing the function's value at that single point.

A very common AP exam question asks: 'For what value of $c$ is the function continuous at $x=a$?' where the function is defined as a piecewise function with a constant at $x=a$. This is exactly a removing discontinuities question: the answer is $c$ equal to the limit of the function as $x \to a$. The entire process only changes the value of the function at one point, leaving all other values unchanged.

**Worked example:** For what value of $c$ is the function $f(x)$ continuous at $x=2$? $f(x) = \begin{cases} \frac{3x^2 - 5x - 2}{x - 2} & x \neq 2 \\ c & x = 2 \end{cases}$

1. 1. By definition, $f(x)$ is continuous at $x=2$ if and only if $\lim_{x \to 2}f(x) = f(2) = c$.
2. 2. Factor the numerator: $3x^2 - 5x - 2 = (3x + 1)(x - 2)$. For $x \neq 2$, we can cancel the common $(x-2)$ factor, so $f(x) = 3x + 1$ for $x \neq 2$.
3. 3. Evaluate the limit: $\lim_{x \to 2} (3x + 1) = 3(2) + 1 = 7$.
4. 4. To remove the discontinuity, set $c = 7$, which makes $f(x)$ continuous at $x=2$.

> **tip**
>
> Never try to plug $x=a$ directly into the original undefined expression to get $c$, this will always give you 0/0 which is undefined. Always calculate the limit first, then set $c$ equal to that limit.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** After canceling a common factor $(x-a)$, claiming the simplified function is equal to the original function everywhere, including at $x=a$.
  - Why it fails: Students forget that canceling removes the restriction $x \neq a$, which was required for the original function to be defined.
  - Correct: Always note the equality only holds for $x \neq a$, and the original function is still undefined at $x=a$ until you redefine it.
- **Wrong:** Confusing a removable discontinuity at $x=a$ with a vertical asymptote because the denominator is zero at $x=a$.
  - Why it fails: All roots of the denominator produce discontinuities, but not all are asymptotes, and students assume any zero denominator means an asymptote.
  - Correct: Always evaluate the limit at every point of discontinuity to check if it is finite and removable.
- **Wrong:** Forgetting to factor out the negative sign after expanding the conjugate difference of squares, e.g., getting $(x-9)$ instead of $-(x-9)$ for the denominator of $\frac{x-9}{3-\sqrt{x}}$.
  - Why it fails: The difference of squares gives $a^2 - x = -(x-a^2)$, which is easy to miss when expanding quickly.
  - Correct: Always expand the denominator step-by-step and factor out any negative sign explicitly before canceling terms.
- **Wrong:** Claiming all 0/0 indeterminate forms produce a finite limit and are removable discontinuities.
  - Why it fails: Students associate 0/0 with removable discontinuities, but 0/0 forms can simplify to functions with infinite limits (e.g., $\lim_{x \to 0} \frac{x}{x^3}$).
  - Correct: Always simplify fully and confirm the limit is finite before classifying a discontinuity as removable.
- **Wrong:** When redefining the function to remove the discontinuity, changing the value of the function at all points to match the simplified expression.
  - Why it fails: Students think the entire function changes after canceling the common factor.
  - Correct: Only change the value of the function at the single point $x=a$ where the discontinuity occurred; leave all other points unchanged.

## Cheatsheet

| Category | Formula / Rule | Notes |
| --- | --- | --- |
| Removable Discontinuity Definition | Discontinuity at $x=a$ is removable iff $\lim_{x \to a} f(x)$ exists and is finite | $f(a)$ can be undefined or defined incorrectly, this does not change the classification |
| Non-Removable Discontinuity | Any discontinuity where the two-sided limit does not exist (finite) | Includes jump, infinite, and oscillating discontinuities; cannot be removed by redefining $f(a)$ |
| Factoring Method for Limits | If $f(x) = \frac{(x-a)g(x)}{(x-a)h(x)}$, then $\lim_{x \to a}f(x) = \frac{g(a)}{h(a)}$ | Only valid if $h(a) \neq 0$; equality holds for all $x \neq a$ |
| Conjugate Method for Limits | For $\lim_{x \to a} \frac{N(x)}{c - \sqrt{x}}$, multiply numerator and denominator by $c + \sqrt{x}$ | Used for functions with square roots where factoring is not possible |
| Find $c$ for Continuity | If $f(x) = \begin{cases} g(x) & x \neq a \\ c & x=a \end{cases}$, set $c = \lim_{x \to a}g(x)$ | This is the most common AP question type for this topic |
| Remove Discontinuity Rule | New continuous function $g(x) = \begin{cases} f(x) & x \neq a \\ \lim_{x \to a}f(x) & x = a \end{cases}$ | Only changes the value of $f(x)$ at the single point $x=a$ |
| Indeterminate Form Clue | A $0/0$ form when evaluating $f(a)$ indicates a potential removable discontinuity | Always confirm the limit is finite after simplification, not all $0/0$ are removable |

## What's next

Removing discontinuities is a foundational skill for AP Calculus Unit 1 that leads directly to the definition of the derivative, the core concept of all calculus. The derivative is defined as the limit of a difference quotient, which always has a $0/0$ indeterminate form at the point of evaluation, so you need the algebraic techniques you learned here to simplify and evaluate derivative limits. Without mastering removing discontinuities, you will struggle to compute derivatives from first principles, and to distinguish holes from vertical asymptotes when analyzing or graphing functions, a common skill tested across both multiple-choice and free-response questions. This topic also lays the groundwork for indeterminate forms and L'Hospital’s Rule for evaluating more complex limits later in the course.

- [Connecting infinite limits and vertical asymptotes](https://www.owlsprep.com/study/ap-calculus-bc-u1-connecting-infinite-limits-and-vertical/)
- [Connecting limits at infinity and horizontal asymptotes](https://www.owlsprep.com/study/ap-calculus-bc-u1-connecting-limits-at-infinity-and/)
- [Working with the Intermediate Value Theorem (IVT)](https://www.owlsprep.com/study/ap-calculus-bc-u1-working-with-the-intermediate-value/)

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