# Can rational functions inside logarithms have removable discontinuities?

• Qube
In summary, the conversation discusses the concept of removable discontinuities and whether or not x = 0 is a removable discontinuity for the function f(x) = ln[(x-x^2)/x]. The conversation also mentions that the factor (x+2) can be canceled out in rational functions, resulting in a removable discontinuity at x = -2. However, it is unclear if the same concept applies to rational functions inside logarithms. The question is posed to the reader to consider whether x = 0 is a removable discontinuity and to explain why or why not.
Qube
Gold Member

## Homework Statement

f(x) = ln[(x-x^2)/x]

Is x = 0 a removable discontinuity?

## Homework Equations

Removable discontinuities are points that can be filled in on a graph to make it continuous.

## The Attempt at a Solution

Is it? I know that with rational functions, canceling out factors can result in removable discontinuities. For example, the function (x+2)/[(x+2)(x+3)] has a removable discontinuity at x = -2 since the factor (x+2) can be canceled out.

What about rational functions inside logarithms?

Qube said:

## Homework Statement

f(x) = ln[(x-x^2)/x]

Is x = 0 a removable discontinuity?

## Homework Equations

Removable discontinuities are points that can be filled in on a graph to make it continuous.

## The Attempt at a Solution

Is it? I know that with rational functions, canceling out factors can result in removable discontinuities. For example, the function (x+2)/[(x+2)(x+3)] has a removable discontinuity at x = -2 since the factor (x+2) can be canceled out.

What about rational functions inside logarithms?

What do you think? If you think the answer is NO, why would you think that? Ditto if you think the answer is YES.

@Qube: you appear to have answered your own question without realizing it.
Probably you need to focus on what it means for a discontinuity to be "removeable".

Removable discontinuities are points that can be filled in on a graph to make it continuous.
... not quite right is it? If you plotted y=(x+2)/[(x+2)(x+3)] would there be a discontinuity on the graph?

## What are removable discontinuities?

Removable discontinuities, also known as removable singularities, are points on a graph where the function is undefined but can be made continuous by redefining the function at that point.

## What causes a removable discontinuity?

A removable discontinuity is caused by a hole or gap in the graph of a function. This can happen when there is a point where the function is undefined, such as when there is a division by zero, but the limit of the function exists.

## How can you determine if a point is a removable discontinuity?

To determine if a point is a removable discontinuity, you can evaluate the limit of the function at that point. If the limit exists and is finite, then the point is a removable discontinuity. If the limit does not exist or is infinite, then the point is not a removable discontinuity.

## Can a removable discontinuity be removed?

Yes, a removable discontinuity can be removed by redefining the function at that point. This can be done by filling in the gap or hole in the graph with the value of the limit at that point. After redefining the function, the point will no longer be a discontinuity and the function will be continuous.

## What is the difference between a removable discontinuity and a non-removable discontinuity?

The main difference between a removable discontinuity and a non-removable discontinuity is that a removable discontinuity can be removed by redefining the function at that point, while a non-removable discontinuity cannot. Non-removable discontinuities, also known as essential singularities, occur when the limit of the function does not exist or is infinite at a point.

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