Which of the functions graphed below has a removable discontinuity – Removable discontinuities, a fascinating concept in the realm of functions, arise when a function exhibits a break in its graph that can be mended by redefining the function at the point of discontinuity. Delving into the intricacies of this mathematical phenomenon, we embark on a journey to uncover the conditions, types, and applications of removable discontinuities, shedding light on their significance in modeling real-world scenarios.

Functions with Removable Discontinuities

In mathematics, a removable discontinuity occurs when a function is discontinuous at a point but can be made continuous by redefining the function at that point. This means that the function has a “hole” or “jump” at that point, but the hole can be “filled in” by assigning a specific value to the function at that point.

For a function to have a removable discontinuity at a point, the following conditions must be met:

1. The function must be undefined at the point.
2. The limit of the function as it approaches the point from both sides must exist and be equal.

Types of Functions with Removable Discontinuities

There are two main types of functions that can have removable discontinuities:

• Functions with a jump discontinuity: These functions have a discontinuity where the function jumps from one value to another.
• Functions with an infinite discontinuity: These functions have a discontinuity where the function approaches infinity as it approaches the point.

Identifying Removable Discontinuities, Which of the functions graphed below has a removable discontinuity

To identify removable discontinuities in a function, follow these steps:

1. Find the points where the function is undefined.
2. Calculate the limit of the function as it approaches the point from both sides.
3. If the limits exist and are equal, then the discontinuity is removable.

Examples

Here is a table of functions with removable discontinuities:

Function	Point of Discontinuity	Limit
f(x) = (x-1)/(x-2)	x = 2	1
f(x) = 1/(x^2-1)	x = 1, x = 1	0
f(x) = |x-1|	x = 1	0

Applications

Removable discontinuities have practical applications in various fields, such as:

• Physics:Modeling the behavior of materials that undergo phase transitions, such as melting or freezing.
• Economics:Modeling the supply and demand of goods and services, which may experience sudden changes due to external factors.
• Computer science:Handling data with missing or incomplete values, where the discontinuity can be removed by imputing a suitable value.

Expert Answers: Which Of The Functions Graphed Below Has A Removable Discontinuity

What are the necessary conditions for a function to have a removable discontinuity?

A function must have a hole in its graph and the limit of the function at that point must exist.

How can you identify a removable discontinuity using the limit definition?

If the left-hand limit and the right-hand limit of the function at a point are equal and finite, then the discontinuity is removable.

What are some examples of functions with removable discontinuities?

Functions with holes, such as f(x) = (x-1)/(x-2), and functions with jump discontinuities, such as f(x) = |x|, can have removable discontinuities.