Discontinuity Analysis Of F(x) = (x³ + X) / X At X = 0

Introduction: Unveiling Discontinuities in Functions

In the realm of mathematics, understanding the behavior of functions is paramount, especially when it comes to identifying points of discontinuity. Discontinuities are points where a function's graph has a break, jump, or hole, and they can significantly impact the function's properties and applications. This article delves into a specific function, f(x) = (x³ + x) / x, to determine whether it exhibits a discontinuity at the point x = 0. We will explore the concept of discontinuities, the criteria for their existence, and the techniques for analyzing a function's behavior around potential points of discontinuity. By meticulously examining the function f(x), we aim to provide a clear and comprehensive answer to the question of its continuity at x = 0. Understanding discontinuities is crucial for various mathematical and scientific applications, including calculus, differential equations, and signal processing. A thorough understanding of these concepts allows us to model real-world phenomena more accurately and solve complex problems involving functions and their behaviors.

Before we dive into the specifics of our function, let's define what we mean by a discontinuity. A function is said to be continuous at a point if its limit exists at that point, the function is defined at that point, and the limit is equal to the function's value at that point. If any of these conditions are not met, the function is discontinuous at that point. There are several types of discontinuities, including removable discontinuities, jump discontinuities, and infinite discontinuities. A removable discontinuity, also known as a hole, occurs when the limit of the function exists at the point, but the function is either not defined or the function's value does not match the limit. A jump discontinuity occurs when the left-hand limit and the right-hand limit exist but are not equal. An infinite discontinuity occurs when the function approaches infinity (or negative infinity) as x approaches the point of discontinuity. In this article, we will specifically investigate if f(x) = (x³ + x) / x possesses a removable discontinuity at x = 0. This type of discontinuity is particularly interesting because it can often be "removed" by redefining the function at that point, making the function continuous. The identification and classification of discontinuities are essential steps in analyzing a function's behavior and its suitability for various applications. We will employ algebraic manipulation and limit analysis to rigorously determine the nature of the function's behavior at x = 0, thereby providing a definitive answer to our central question.

Analyzing f(x) = (x³ + x) / x for Discontinuity at x = 0

To determine if the function f(x) = (x³ + x) / x has a discontinuity at x = 0, we need to analyze its behavior around this point. The first step in this analysis involves simplifying the function algebraically. We can factor out an x from the numerator, resulting in f(x) = x(x² + 1) / x. Notice that for all x not equal to 0, we can cancel the x terms, simplifying the function to f(x) = x² + 1. This simplified form is crucial because it reveals the underlying behavior of the function away from the potential point of discontinuity. The simplification highlights that the function behaves like a parabola, specifically x² + 1, everywhere except possibly at x = 0. At x = 0, the original function is undefined because we would be dividing by zero. This initial observation suggests that there might be a discontinuity at x = 0, but we need to investigate further to determine the type of discontinuity.

Now, let's examine the limit of the function as x approaches 0. We are interested in determining if the function approaches a specific value as x gets closer and closer to 0. Since we have simplified the function to f(x) = x² + 1 for all x not equal to 0, we can evaluate the limit of this simplified expression as x approaches 0. Mathematically, we write this as lim (x→0) (x² + 1). Substituting x = 0 into the simplified expression, we get 0² + 1 = 1. This means that as x approaches 0, the function f(x) approaches the value 1. The existence of this limit is a key piece of information in determining the nature of the discontinuity, if any. The fact that the limit exists suggests that the discontinuity might be a removable discontinuity. To confirm this, we need to compare the limit to the function's value at x = 0. However, as we noted earlier, the original function is undefined at x = 0, which further supports the hypothesis of a removable discontinuity. This analysis of the limit is crucial for understanding the function's behavior in the vicinity of the point of interest. By determining the limit, we gain insight into the function's trend and whether it approaches a specific value as we get closer to the potential discontinuity.

Identifying and Classifying the Discontinuity

Having established that the limit of f(x) as x approaches 0 is 1, and that f(x) is undefined at x = 0, we can confidently conclude that the function has a removable discontinuity at x = 0. A removable discontinuity, often referred to as a hole, occurs when the limit of the function exists at a point, but the function is either not defined at that point or its value at that point differs from the limit. In this case, the limit exists and is equal to 1, but the function f(x) = (x³ + x) / x is undefined at x = 0 due to the division by zero. This perfectly fits the definition of a removable discontinuity. The discontinuity is