Removable Discontinuity Of F(x) = (x^2 + 3x - 70) / (x + 10) A Detailed Analysis

In the fascinating realm of calculus and mathematical analysis, the concept of discontinuity plays a pivotal role in understanding the behavior of functions. A discontinuity, simply put, is a point where a function's graph is not continuous – it has a break, a jump, or a hole. These discontinuities can manifest in various forms, each with its unique characteristics and implications. Among these, the removable discontinuity stands out as a particularly intriguing case.

Understanding Removable Discontinuities

A removable discontinuity, as the name suggests, is a point where a function is discontinuous, but this discontinuity can be 'removed' by redefining the function at that specific point. In essence, it's a point where the function has a hole, but if we were to fill that hole, the function would become continuous at that point. This occurs when the limit of the function exists at that point, but the function is either undefined or its value differs from the limit. To identify a removable discontinuity, we often look for factors that cancel out in the numerator and denominator of a rational function. This cancellation creates a hole in the graph, which is the removable discontinuity.

Delving into the Function f(x) = (x^2 + 3x - 70) / (x + 10)

Let's turn our attention to the specific function at hand: f(x) = (x^2 + 3x - 70) / (x + 10). This is a rational function, a ratio of two polynomials, which are notorious for exhibiting discontinuities. To unravel the nature of this function's discontinuities, we embark on a journey of algebraic manipulation and limit evaluation. The key lies in factoring the numerator and examining potential cancellations with the denominator. Factoring the quadratic expression in the numerator, we seek two numbers that multiply to -70 and add up to 3. These numbers are 10 and -7. Hence, we can rewrite the numerator as (x + 10)(x - 7). Our function now transforms into f(x) = ((x + 10)(x - 7)) / (x + 10). A crucial observation arises: the factor (x + 10) appears in both the numerator and the denominator. This immediately signals the potential presence of a removable discontinuity. The cancellation of this common factor is the hallmark of a removable discontinuity, signifying a hole in the graph at the x-value that makes this factor zero.

Identifying and Resolving the Discontinuity

The Critical Point: x = -10

The factor (x + 10) becomes zero when x = -10. This is the critical point where our function might exhibit a discontinuity. At x = -10, the denominator of the original function becomes zero, rendering the function undefined. This confirms our suspicion of a discontinuity at this point. However, the fact that we could cancel this factor suggests that it's a removable discontinuity. To solidify our understanding, we simplify the function by canceling the (x + 10) terms, resulting in a new function: g(x) = x - 7, for x ≠ -10. This simplified function is identical to the original function everywhere except at x = -10. It represents the 'repaired' version of the function, the one where the hole has been filled.

The Limit Reveals the Truth

To formally confirm the removable discontinuity, we evaluate the limit of the function as x approaches -10. The limit, in essence, tells us where the function is 'heading' as it gets arbitrarily close to a particular point. Using the simplified function, we find: lim (x→-10) g(x) = lim (x→-10) (x - 7) = -10 - 7 = -17. The limit exists and is equal to -17. This signifies that as x approaches -10, the function approaches -17. However, the original function f(x) is undefined at x = -10. This discrepancy between the limit and the function's value (or lack thereof) at the point is the defining characteristic of a removable discontinuity. The graph of f(x) has a hole at the point (-10, -17).

Removing the Discontinuity: Redefining the Function

Now comes the 'removal' part. We can create a new function, let's call it h(x), which is identical to f(x) everywhere except at x = -10, and is defined to be equal to the limit at x = -10. This is how we 'fill the hole'. h(x) = { (x^2 + 3x - 70) / (x + 10), if x ≠ -10; -17, if x = -10 }. Or, more simply, h(x) = { x - 7, if x ≠ -10; -17, if x = -10 }. This new function, h(x), is continuous everywhere, including at x = -10. We have successfully removed the discontinuity.

A graphical representation provides a compelling visual confirmation of the removable discontinuity. The graph of f(x) will appear as a straight line, identical to the graph of g(x) = x - 7, but with a tiny hole at the point (-10, -17). This hole represents the point where the function is undefined. When we graph the redefined function h(x), the hole is filled, and we see a continuous line. This visual contrast vividly illustrates the nature of a removable discontinuity and how it can be 'removed'. Graphing the function using software like Desmos or Geogebra can provide a clear picture of this phenomenon.

Removable discontinuities are not just mathematical curiosities; they have significant implications in various areas of calculus and its applications. In the context of limits, they highlight the difference between the value a function approaches and the actual value of the function at a point. In the study of continuity, they emphasize that a function must be both defined and have a limit at a point to be continuous there. Moreover, in applications such as physics and engineering, understanding removable discontinuities is crucial for modeling physical systems accurately. For instance, in circuit analysis, a removable discontinuity might represent a scenario where a component is temporarily disconnected, leading to a momentary interruption in the circuit's behavior.

In conclusion, the function f(x) = (x^2 + 3x - 70) / (x + 10) elegantly demonstrates the concept of a removable discontinuity. By factoring, simplifying, and evaluating limits, we identified a hole at x = -10. We then successfully 'removed' the discontinuity by redefining the function at that point. This exploration not only deepens our understanding of discontinuities but also showcases the power of algebraic manipulation and limit evaluation in unraveling the behavior of functions. Removable discontinuities, while seemingly minor, play a vital role in the broader landscape of calculus and its applications, reminding us that even the smallest 'holes' can have significant implications.

Introduction to Discontinuity

In the world of mathematics, particularly in calculus, discontinuity is a fundamental concept that describes points where a function behaves irregularly. Unlike continuous functions, which can be drawn without lifting a pen from the paper, discontinuous functions exhibit breaks, jumps, or holes at certain points. Understanding discontinuities is crucial for analyzing the behavior of functions and their applications in various fields. Among the different types of discontinuities, the removable discontinuity is a unique case that we will explore in detail.

Deep Dive into Removable Discontinuity

A removable discontinuity occurs at a point where a function is undefined, but the limit of the function exists at that point. This means that there is a 'hole' in the graph of the function, but if we were to 'fill' that hole, the function would become continuous at that point. This type of discontinuity is often associated with rational functions, where factors in the numerator and denominator can be canceled out. When a factor cancels out, it indicates a removable discontinuity at the x-value that makes that factor equal to zero. This is because the cancellation implies that the function behaves smoothly near that point, except for a single point where it is undefined.

Identifying Removable Discontinuity: A Step-by-Step Approach

Identifying a removable discontinuity involves a systematic process of analyzing the function. The first step is to look for potential points of discontinuity, which often occur where the denominator of a rational function equals zero. Once these points are identified, the next step is to simplify the function by factoring and canceling out common factors in the numerator and denominator. If a factor cancels out, it indicates a removable discontinuity at the x-value that makes that factor zero. To confirm the discontinuity, we need to evaluate the limit of the function as x approaches the point of discontinuity. If the limit exists, then it is indeed a removable discontinuity. Finally, we can 'remove' the discontinuity by redefining the function at that point to be equal to the limit.

Detailed Example: f(x) = (x^2 + 3x - 70) / (x + 10) Revisited

Let's revisit the function f(x) = (x^2 + 3x - 70) / (x + 10) to illustrate the process of identifying and handling a removable discontinuity. The first step is to identify potential points of discontinuity. The denominator, (x + 10), becomes zero when x = -10. So, x = -10 is a potential point of discontinuity. Next, we simplify the function by factoring the numerator: f(x) = ((x + 10)(x - 7)) / (x + 10). We notice that the factor (x + 10) appears in both the numerator and the denominator, indicating a removable discontinuity. Canceling out the common factor, we get the simplified function g(x) = x - 7, for x ≠ -10. To confirm the removable discontinuity, we evaluate the limit of f(x) as x approaches -10: lim (x→-10) f(x) = lim (x→-10) (x - 7) = -17. The limit exists and is equal to -17. This confirms that there is a removable discontinuity at x = -10. To remove the discontinuity, we redefine the function at x = -10 to be equal to the limit: h(x) = { (x^2 + 3x - 70) / (x + 10), if x ≠ -10; -17, if x = -10 }. This redefined function, h(x), is continuous everywhere.

Graphical Insights: Visualizing Removable Discontinuities

The graph of a function with a removable discontinuity provides a clear visual representation of the concept. The graph will appear smooth and continuous everywhere except at the point of discontinuity, where there will be a 'hole'. This hole represents the point where the function is undefined. However, the function approaches a specific value as x approaches the point of discontinuity, which is the limit we calculated earlier. When we redefine the function to 'fill' the hole, the graph becomes completely continuous. Visualizing these graphs helps in understanding the nature of removable discontinuities and their impact on the behavior of functions. Tools like Desmos and Geogebra are invaluable for plotting these functions and observing the discontinuities.

Real-World Applications and Significance

Removable discontinuities are not just theoretical concepts; they have practical applications in various fields. In physics, they can represent situations where a system undergoes a temporary interruption or a sudden change that can be 'removed' by redefining certain parameters. In engineering, they can arise in the analysis of circuits or control systems, where components may be temporarily disconnected. Understanding removable discontinuities allows engineers to model these systems accurately and design them to handle such situations. Moreover, in computer graphics and image processing, removable discontinuities can appear in functions that define shapes or textures, and proper handling of these discontinuities is crucial for rendering smooth and realistic images. The ability to identify and remove these discontinuities ensures that the mathematical models accurately represent the physical phenomena they describe, leading to more reliable predictions and designs.

Conclusion: The Elegance of Removable Discontinuities

In conclusion, removable discontinuities are a fascinating aspect of function analysis. They highlight the nuances of continuity and the importance of limits in understanding function behavior. By systematically identifying potential discontinuities, simplifying functions, and evaluating limits, we can effectively handle removable discontinuities and redefine functions to achieve continuity. This understanding is not only crucial for mathematical analysis but also has significant implications in various real-world applications. The concept of removable discontinuity underscores the elegance and power of calculus in describing and modeling the world around us. Mastering these concepts equips us with the tools to analyze complex systems and make informed decisions based on mathematical insights.

Introduction to Problem-Solving

Understanding the theory behind removable discontinuities is essential, but the true test of knowledge lies in applying these concepts to solve problems. This section focuses on practical examples that illustrate how to identify, analyze, and address removable discontinuities in various functions. By working through these examples, we will solidify our understanding and develop the problem-solving skills necessary to tackle more complex scenarios. Each example will walk through the steps involved in finding the discontinuity, evaluating the limit, and redefining the function to remove the discontinuity.

Example 1: A Classic Rational Function

Consider the function f(x) = (x^2 - 4) / (x - 2). Our goal is to determine if this function has a removable discontinuity and, if so, to remove it. The first step is to identify potential points of discontinuity. The denominator, (x - 2), becomes zero when x = 2. So, x = 2 is a potential point of discontinuity. Next, we simplify the function by factoring the numerator: f(x) = ((x + 2)(x - 2)) / (x - 2). We observe that the factor (x - 2) appears in both the numerator and the denominator, indicating a removable discontinuity. Canceling out the common factor, we obtain the simplified function g(x) = x + 2, for x ≠ 2. To confirm the removable discontinuity, we evaluate the limit of f(x) as x approaches 2: lim (x→2) f(x) = lim (x→2) (x + 2) = 4. The limit exists and is equal to 4. This confirms that there is a removable discontinuity at x = 2. To remove the discontinuity, we redefine the function at x = 2 to be equal to the limit: h(x) = { (x^2 - 4) / (x - 2), if x ≠ 2; 4, if x = 2 }. The redefined function, h(x), is continuous everywhere. This example demonstrates the classic approach to handling removable discontinuities in rational functions: factoring, canceling, evaluating limits, and redefining the function.

Example 2: A Trigonometric Twist

Now, let's consider a function with a trigonometric component: f(x) = (sin(x)) / x. This function is undefined at x = 0, as division by zero is not allowed. To determine if this is a removable discontinuity, we need to evaluate the limit of f(x) as x approaches 0. This limit is a well-known result in calculus: lim (x→0) (sin(x)) / x = 1. The limit exists and is equal to 1. Therefore, there is a removable discontinuity at x = 0. To remove this discontinuity, we redefine the function at x = 0 to be equal to the limit: h(x) = { (sin(x)) / x, if x ≠ 0; 1, if x = 0 }. This redefined function, h(x), is continuous everywhere. This example highlights that removable discontinuities can also occur in functions involving trigonometric expressions and that limit evaluation is crucial in identifying and addressing them.

Example 3: A Piecewise Function

Consider the piecewise function f(x) = { x^2, if x < 1; 2, if x = 1; 2x, if x > 1 }. We need to analyze the continuity of this function at x = 1. To do this, we evaluate the left-hand limit, the right-hand limit, and the function's value at x = 1. The left-hand limit is lim (x→1-) f(x) = lim (x→1-) x^2 = 1. The right-hand limit is lim (x→1+) f(x) = lim (x→1+) 2x = 2. The function's value at x = 1 is f(1) = 2. Since the left-hand limit is not equal to the right-hand limit, the limit lim (x→1) f(x) does not exist. Therefore, the function has a discontinuity at x = 1. However, it's not a removable discontinuity in the traditional sense, as the limit does not exist. To make the function continuous, we would need to redefine the function not only at x = 1 but also adjust the definition for either x < 1 or x > 1 to make the limits match. This example illustrates that not all discontinuities are removable, and a careful analysis of limits is necessary to determine the type of discontinuity.

Importance of Practice and Visualization

Solving problems involving removable discontinuities requires a combination of algebraic manipulation, limit evaluation, and careful analysis. Practice is key to mastering these skills. Working through a variety of examples, including rational, trigonometric, and piecewise functions, helps to develop a strong intuition for identifying and addressing removable discontinuities. Visualization is also a valuable tool. Graphing the functions using software like Desmos or Geogebra can provide a clear picture of the discontinuity and the effect of redefining the function to remove it. Visualizing the hole in the graph and how it is 'filled' by redefining the function reinforces the concept and makes it more intuitive. Combining practice with visualization is the most effective way to build a solid understanding of removable discontinuities and their applications.

Conclusion: Mastering the Art of Solving Problems

In conclusion, solving problems involving removable discontinuities is a crucial aspect of mastering this concept. The examples discussed here demonstrate the systematic approach to identifying, analyzing, and addressing these discontinuities in various types of functions. By practicing these techniques and visualizing the results, we can develop the skills necessary to tackle more complex problems and apply this knowledge in real-world scenarios. The ability to handle removable discontinuities effectively is a testament to a deep understanding of calculus and its applications. These skills are invaluable in various fields, from engineering and physics to computer graphics and data analysis, where understanding function behavior is essential for accurate modeling and decision-making.