Removable Discontinuity In F(x)=(x^2-49)/(x+7) A Detailed Analysis

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In the realm of mathematical analysis, understanding the behavior of functions is paramount. A key aspect of this understanding lies in identifying and classifying discontinuities. Discontinuities are points where a function is not continuous, meaning there's a break or interruption in the graph. These discontinuities can manifest in various forms, each with its unique characteristics and implications. One particularly interesting type of discontinuity is the removable discontinuity. In this article, we delve deep into the concept of removable discontinuities, focusing on the function $f(x)=\frac{x^2-49}{x+7}$. We will explore what removable discontinuities are, how to identify them, and the significance they hold in the broader context of calculus and function analysis. By the end of this exploration, you will have a comprehensive understanding of how to analyze functions for removable discontinuities and the steps involved in "removing" them to create a continuous function.

What is Removable Discontinuity?

A removable discontinuity occurs at a point where a function is undefined, but the limit of the function exists at that point. In simpler terms, there's a "hole" in the graph of the function, but if you could just fill that hole, the function would be continuous at that point. This is in contrast to other types of discontinuities, such as infinite discontinuities (where the function approaches infinity) or jump discontinuities (where the function abruptly jumps from one value to another), which cannot be "fixed" by simply redefining the function at a single point. The essence of a removable discontinuity is that the discontinuity can be eliminated by redefining the function at that specific point, making the function continuous there. This unique characteristic makes them particularly interesting and relevant in various mathematical contexts, such as simplifying functions, evaluating limits, and understanding the behavior of complex functions. Understanding removable discontinuities is crucial for advanced calculus and real analysis concepts. It allows mathematicians and students to manipulate functions in ways that preserve their overall behavior while eliminating problematic points.

Key Characteristics of Removable Discontinuities

  • Limit Existence: The most crucial characteristic of a removable discontinuity is that the limit of the function exists at the point of discontinuity. This means that as x approaches the value where the function is discontinuous, the function approaches a specific value from both the left and the right. This convergence is what allows us to "fill" the hole and make the function continuous.
  • Undefined Function Value: At the point of discontinuity, the function itself is undefined. This typically occurs due to division by zero or other indeterminate forms. For instance, in the function $f(x)=\frac{x^2-49}{x+7}$, the function is undefined at x = -7 because the denominator becomes zero.
  • Redefinition for Continuity: The discontinuity can be "removed" by redefining the function at the point of discontinuity. This involves assigning the limit value to the function at that point, effectively filling the hole and making the function continuous. For example, if the limit of f(x) as x approaches a is L, we can redefine the function such that f(a) = L, thus removing the discontinuity.

Identifying Removable Discontinuity in $f(x)=\frac{x^2-49}{x+7}$

To identify a removable discontinuity in the given function $f(x)=\frac{x^2-49}{x+7}$, we need to follow a systematic approach. This involves analyzing the function for points where it is undefined and then examining the limit of the function at those points. The process typically involves factoring, simplifying, and evaluating the limit to determine if a removable discontinuity exists. This methodical approach is essential for accurately identifying and characterizing discontinuities in functions. Without a structured method, it can be challenging to distinguish between different types of discontinuities and to understand their implications. In this section, we will break down the process into clear, manageable steps, providing a detailed explanation of each stage. This will not only help in identifying the removable discontinuity in the given function but also equip you with a general strategy for analyzing other functions in the future.

Step-by-Step Identification Process

  1. Identify Points of Undefined Behavior: The first step in identifying a removable discontinuity is to find the points where the function is undefined. This typically occurs when the denominator of a rational function is equal to zero. In our function, $f(x)=\frac{x^2-49}{x+7}$, the denominator is x + 7. Setting this equal to zero gives us x + 7 = 0, which implies x = -7. Therefore, the function is undefined at x = -7. This point is a potential candidate for a discontinuity, and we need to investigate further.
  2. Simplify the Function: The next step is to simplify the function, if possible. This often involves factoring the numerator and denominator to see if any common factors can be canceled out. In our case, the numerator is a difference of squares, which can be factored as x2 - 49 = (x - 7)(x + 7). So, our function becomes:

    f(x)=(xβˆ’7)(x+7)x+7f(x)=\frac{(x-7)(x+7)}{x+7}

    We can see that (x + 7) is a common factor in both the numerator and the denominator. Canceling out this common factor (with the condition that x β‰  -7) simplifies the function to:

    f(x)=xβˆ’7,xβ‰ βˆ’7f(x) = x - 7, \quad x \neq -7

    This simplification is a crucial step because it reveals the underlying behavior of the function. The simplified form makes it clear that the function behaves like a linear function x - 7, except at the point x = -7.
  3. Evaluate the Limit: Now, we need to evaluate the limit of the simplified function as x approaches the point of undefined behavior, which is x = -7. This will tell us whether the function approaches a specific value as we get closer to this point. The limit is:

    lim⁑xβ†’βˆ’7(xβˆ’7)\lim_{x \to -7} (x - 7)

    Substituting x = -7 into the simplified function gives us:

    lim⁑xβ†’βˆ’7(xβˆ’7)=βˆ’7βˆ’7=βˆ’14\lim_{x \to -7} (x - 7) = -7 - 7 = -14

    The limit exists and is equal to -14. This is a key indication of a removable discontinuity. The existence of a finite limit suggests that we can redefine the function at x = -7 to make it continuous.
  4. Confirm Removable Discontinuity: Since the limit exists at x = -7, we can confirm that there is a removable discontinuity at this point. The function is undefined at x = -7, but the limit as x approaches -7 is -14. This means that if we were to fill the "hole" in the graph at x = -7 with the value -14, the function would be continuous at that point.

Graphical Representation of the Discontinuity

Visualizing the function $f(x)=\frac{x^2-49}{x+7}$ graphically provides a clear understanding of the removable discontinuity. When graphed, the function appears as a straight line, specifically the line y = x - 7. However, there is a notable exception: at the point x = -7, there is a hole in the graph. This hole represents the point where the function is undefined due to the denominator becoming zero. The graph visually confirms the existence of a removable discontinuity. The straight line indicates the function's behavior in general, while the hole marks the specific point where the function deviates from this behavior.

Interpreting the Graph

  • The Hole: The hole at x = -7 signifies that the function does not have a defined value at this point. If you were to trace the graph with your finger, you would have to lift it at x = -7 and then continue on the other side. This break in the graph is a visual representation of the discontinuity.
  • The Line: Away from x = -7, the graph is a straight line, which is the graph of the simplified function y = x - 7. This shows that the function behaves linearly except at the point of discontinuity.
  • The Limit: The fact that the graph approaches a specific y-value as x approaches -7 from both sides visually demonstrates the existence of the limit. This is crucial for identifying the discontinuity as removable rather than another type of discontinuity.

Creating the Graph

To create the graph of $f(x)=\frac{x^2-49}{x+7}$, you can use graphing software or plot points manually. Here’s a general approach:

  1. Simplify the Function: First, simplify the function to f(x) = x - 7, keeping in mind that x β‰  -7.
  2. Plot the Line: Graph the line y = x - 7. This is a straight line with a slope of 1 and a y-intercept of -7.
  3. Indicate the Hole: At x = -7, there should be a hole. This is typically represented as an open circle on the graph. The y-coordinate of the hole is the limit value, which we found to be -14. So, the hole is at the point (-7, -14).

Removing the Discontinuity: Redefining the Function

Once we have identified a removable discontinuity, the next step is to remove it by redefining the function. This process involves assigning a value to the function at the point of discontinuity, effectively "filling" the hole in the graph. The value we assign is the limit of the function as x approaches that point. This technique is fundamental in calculus and analysis as it allows us to work with functions as if they were continuous, even when they have isolated points of discontinuity. By redefining the function, we create a new function that is identical to the original everywhere except at the point of discontinuity, where it is now continuous. This redefined function is often more convenient to work with, especially when performing operations like differentiation or integration.

The Process of Redefinition

  1. Determine the Limit: As we established earlier, the first step in removing the discontinuity is to find the limit of the function as x approaches the point of discontinuity. For our function, $f(x)=\frac{x^2-49}{x+7}$, the point of discontinuity is x = -7, and the limit as x approaches -7 is -14.
  2. Redefine the Function: Now, we redefine the function by assigning the limit value to the function at the point of discontinuity. We create a piecewise function that matches the original function everywhere except at the point of discontinuity, where it takes on the limit value. The redefined function, g(x), is as follows:

    g(x)={x2βˆ’49x+7,ifΒ xβ‰ βˆ’7βˆ’14,ifΒ x=βˆ’7 g(x) = \begin{cases} \frac{x^2-49}{x+7}, & \text{if } x \neq -7 \\ -14, & \text{if } x = -7 \end{cases}

    This piecewise function is identical to the original function for all x except x = -7. At x = -7, the redefined function is explicitly defined as -14, which is the limit of the original function as x approaches -7.
  3. Verify Continuity: The final step is to verify that the redefined function is indeed continuous at the point where the original function was discontinuous. A function is continuous at a point if the limit of the function as x approaches that point is equal to the function's value at that point. In our case, we need to check that:

    lim⁑xβ†’βˆ’7g(x)=g(βˆ’7)\lim_{x \to -7} g(x) = g(-7)

    We know that g(-7) = -14 by definition. We also know that the limit of g(x) as x approaches -7 is the same as the limit of the original function, which is -14. Therefore,

    lim⁑xβ†’βˆ’7g(x)=βˆ’14=g(βˆ’7)\lim_{x \to -7} g(x) = -14 = g(-7)

    This confirms that the redefined function g(x) is continuous at x = -7, and we have successfully removed the discontinuity.

Implications and Applications

Understanding and handling removable discontinuities is not just an academic exercise; it has significant implications and applications in various areas of mathematics and beyond. Removable discontinuities can affect the behavior of functions in calculus, real analysis, and other fields, making it crucial to identify and address them. The ability to redefine functions to remove discontinuities allows for smoother and more straightforward analysis, particularly when dealing with integrals, derivatives, and other advanced concepts. Moreover, the concept extends beyond pure mathematics, finding relevance in engineering, physics, and computer science, where functions are used to model real-world phenomena.

Applications in Calculus

  • Evaluating Limits: Removable discontinuities are often encountered when evaluating limits. By identifying and removing the discontinuity, we can directly substitute the value into the simplified function to find the limit, as we did in our example. This simplifies the limit evaluation process.
  • Differentiation: Differentiability requires continuity. If a function has a non-removable discontinuity, it is not differentiable at that point. However, if the discontinuity is removable, we can redefine the function to make it continuous and then differentiate it. This is particularly useful in optimization problems and curve sketching.
  • Integration: Similarly, for integration, continuity is desirable. Removable discontinuities can be addressed by redefining the function, making integration simpler. In definite integrals, removing the discontinuity can allow us to apply standard integration techniques more easily.

Applications in Real Analysis

  • Continuity and Convergence: In real analysis, the concept of continuity is fundamental. Removable discontinuities provide a context for understanding how functions can be modified to ensure continuity, which is vital for proving convergence theorems and other advanced results.
  • Function Spaces: The space of continuous functions is often used in analysis. By understanding how to remove discontinuities, we can map discontinuous functions to continuous ones, facilitating the use of powerful analytical tools.

Practical Applications

  • Signal Processing: In signal processing, signals can sometimes have discontinuities due to noise or other factors. Identifying and removing these discontinuities is crucial for accurate signal analysis and reconstruction.
  • Control Systems: Control systems often involve functions that model the behavior of physical systems. Removable discontinuities in these functions can represent minor imperfections or idealizations that can be addressed to improve system performance.
  • Computer Graphics: In computer graphics, functions are used to define shapes and surfaces. Removable discontinuities can cause rendering artifacts, and removing them can lead to smoother and more visually appealing graphics.

In summary, removable discontinuities are a fascinating and important aspect of function analysis. They represent points where a function is undefined, but the limit exists, allowing us to redefine the function to achieve continuity. We have explored the concept of removable discontinuities through the example of the function $f(x)=\frac{x^2-49}{x+7}$, demonstrating how to identify, visualize, and remove such discontinuities. The process involves identifying points of undefined behavior, simplifying the function, evaluating the limit, and redefining the function using a piecewise approach. This exploration not only provides a clear understanding of removable discontinuities but also highlights their significance in calculus, real analysis, and various practical applications. The ability to recognize and handle removable discontinuities is a valuable skill for anyone working with functions, whether in mathematics, engineering, or other fields. By mastering this concept, one can gain a deeper appreciation for the nuances of function behavior and the power of mathematical analysis.