Study Notes on Removable Discontinuity

Table of Contents

1. Introduction to Removable Discontinuity
2. Definition and Concept
3. Identifying Removable Discontinuities
4. Removing Removable Discontinuities
5. Worked Example
6. Practice Questions
7. Glossary
8. Summary and Key Takeaways
9. Exam Strategy

Introduction to Removable Discontinuity

Understanding removable discontinuities is crucial for mastering continuity in functions, a fundamental topic in calculus. This guide will help you identify and remove these discontinuities, ensuring a smooth learning experience.

Definition and Concept

What is a Removable Discontinuity?

A removable discontinuity is a type of discontinuity in a function that can be "removed" to make the function continuous over an interval that includes the discontinuity.

- $f(c)$ exists, - $\lim\_{{x \to c}} f(x)$ exists, - and $\lim\_{{x \to c}} f(x) = f(c)$.

Characteristics of Removable Discontinuity

A removable discontinuity is essentially a "hole" in the function. It is a point where:

• $\lim_{{x \to c}} f(x)$ exists,
• but $f(c)$ does not exist.

For instance, consider the function $g(x) = \frac{x}{x}$. This function is not continuous at $x=0$ because $g(0)$ does not exist. However, the discontinuity at $x=0$ is **removable** because $\lim\_{{x \to 0}} g(x) = 1$. The limit exists at the point of discontinuity.

Identifying Removable Discontinuities

To identify a removable discontinuity:

1. Check if $\lim_{{x \to c}} f(x)$ exists.
2. Verify that $f(c)$ does not exist or is not equal to the limit.

Removing Removable Discontinuities

How to Remove a Removable Discontinuity

To remove a removable discontinuity, you need to redefine the function at the point of discontinuity:

1. Define $f(c)$ to be equal to $\lim_{{x \to c}} f(x)$.
2. This redefinition makes the function continuous at that point.

For example, redefine the function $g$ by: $$g(x) = \begin{cases} \frac{x}{x} & x \neq 0 \\\\ 1 & x = 0 \end{cases}$$ This redefinition removes the discontinuity at $x=0$.

Worked Example

Let $f$ be the function defined by $f(x) = \frac{x^2 + 3x}{x}$.

(a) Explain why $f$ is not continuous at $x=0$.

Answer:

At $x=0$: $$f(0) = \frac{0^2 + 3 \cdot 0}{0} = \frac{0}{0}$$

Since $\frac{0}{0}$ is not defined, $f$ is not continuous at $x=0$.

(b) Explain how the discontinuity at $x=0$ can be removed.

Answer:

First, find the limit as $x$ approaches 0 by factorizing and simplifying: $$\frac{x^2 + 3x}{x} = \frac{x(x + 3)}{x} = x + 3$$

Then, $$\lim_{{x \to 0}} f(x) = 0 + 3 = 3$$

The limit exists, so the discontinuity at $x=0$ is removable. Redefine the function so that: $$f(0) = \lim_{{x \to 0}} f(x) = 3$$

Thus, the redefined function is: $$f(x) = \begin{cases} \frac{x^2 + 3x}{x} & x \neq 0 \\ 3 & x = 0 \end{cases}$$

This removes the discontinuity.

Practice Questions

Practice Question

1. Identify the removable discontinuities in the following functions and redefine them to remove the discontinuities:

(a) $h(x) = \frac{x^2 - 1}{x - 1}$

(b) $k(x) = \frac{\sin(x)}{x}$

(c) $m(x) = \frac{x^2 - 4}{x - 2}$

2. Determine if the following function has a removable discontinuity at $x=2$:

$f(x) = \frac{x^2 - 4}{x - 2}$

3. Explain why $g(x) = \frac{x^3 - x}{x}$ is not continuous at $x=0$ and how to remove the discontinuity.

Glossary

• Removable Discontinuity: A discontinuity that can be "removed" by redefining the function at the point of discontinuity.
• Limit: The value that a function approaches as the input approaches some value.
• Continuous Function: A function without any breaks, jumps, or holes in its graph.

Summary and Key Takeaways

Key Points

• A removable discontinuity is a "hole" in a function where the limit exists, but the function value does not.
• To remove a removable discontinuity, redefine the function at the discontinuous point to equal the limit.
• Understanding limits and continuity is crucial for identifying and removing removable discontinuities.

Key Takeaways

• Always check the existence of the limit and the function value at the point of discontinuity.
• Redefine the function to make it continuous at the point of discontinuity.

Exam Strategy

Good luck with your studies, and remember, practice makes perfect!