Identify The Functions That Exhibit A Removable Discontinuity.

Identifying Functions with Removable Discontinuities

Removable discontinuities, also known as holes, represent a specific type of discontinuity in a function where the limit of the function exists at a point, but the function is either undefined or has a different value at that point. Understanding how to identify these discontinuities is crucial in calculus and analysis, offering insights into the function's behavior and allowing for potential "repair" through redefinition. This comprehensive guide will delve into the identification of functions exhibiting removable discontinuities, providing numerous examples and strategies for effective analysis.

Understanding Discontinuities

Before focusing on removable discontinuities specifically, it's essential to establish a firm understanding of discontinuities in general. A discontinuity occurs at a point in the domain of a function where the function is not continuous. Continuity, in simple terms, means that the function's graph can be drawn without lifting the pen. There are three primary types of discontinuities:

1. Removable Discontinuities:

These discontinuities are characterized by the existence of a limit at the point of discontinuity. The function is either undefined or has a different value at this point, creating a "hole" in the graph. These are the focus of this article.

2. Jump Discontinuities:

In jump discontinuities, the limit of the function does not exist at the point of discontinuity because the left-hand limit and the right-hand limit are different. This results in a "jump" in the graph.

3. Infinite Discontinuities:

These discontinuities occur when the function approaches positive or negative infinity as the input approaches the point of discontinuity. The graph typically has a vertical asymptote at these points.

Identifying Removable Discontinuities: A Step-by-Step Approach

Identifying removable discontinuities involves a systematic approach that combines analytical techniques with graphical interpretation. Here's a detailed breakdown:

1. Analyze the Function's Expression:

The first step involves carefully examining the function's algebraic expression. Look for factors that can be canceled out from both the numerator and the denominator. The presence of such common factors often indicates a removable discontinuity.

Example 1:

Consider the function: f(x) = (x² - 4) / (x - 2)

Notice that the numerator can be factored as (x - 2)(x + 2). Thus, we can rewrite the function as:

f(x) = (x - 2)(x + 2) / (x - 2)

For x ≠ 2, we can cancel the (x - 2) terms, leaving f(x) = x + 2. However, at x = 2, the original function is undefined. This indicates a removable discontinuity at x = 2. The limit as x approaches 2 is 4, but f(2) is undefined.

2. Calculate the Limit:

After identifying potential common factors, compute the limit of the simplified function as x approaches the point where the discontinuity is suspected. If the limit exists (i.e., it's a finite value), it confirms a removable discontinuity.

Example 2:

Consider the function: g(x) = (x³ - 8) / (x² - 4)

Factoring the numerator and denominator, we get:

g(x) = (x - 2)(x² + 2x + 4) / (x - 2)(x + 2)

Again, for x ≠ 2, we can cancel (x - 2), leaving g(x) = (x² + 2x + 4) / (x + 2).

Let's evaluate the limit as x approaches 2:

lim (x→2) [(x² + 2x + 4) / (x + 2)] = (4 + 4 + 4) / (4) = 3

The limit exists and equals 3, indicating a removable discontinuity at x = 2.

3. Graph the Function:

Graphing the function can visually confirm the presence of a removable discontinuity. A "hole" in the graph at a specific point signifies a removable discontinuity. Many graphing calculators and software packages can assist in this process. However, it's crucial to understand that the graph might not always perfectly reveal subtle discontinuities. Algebraic analysis remains essential for definitive identification.

4. Piecewise Functions:

Removable discontinuities are frequently encountered in piecewise functions. Careful analysis of the function's definition at the transition points is necessary.

Example 3:

Consider the piecewise function:

h(x) = { x², x < 2 { 4, x = 2 { x + 2, x > 2

The limit as x approaches 2 from the left is lim (x→2⁻) x² = 4. The limit as x approaches 2 from the right is lim (x→2⁺) (x + 2) = 4. Both one-sided limits are equal to 4. However, h(2) = 4. Therefore there is no discontinuity. Let’s change h(2) to 3, then it will have a removable discontinuity at x = 2.

5. Trigonometric Functions:

Removable discontinuities can also arise in trigonometric functions, often involving indeterminate forms. Using L'Hôpital's Rule or trigonometric identities can help in evaluating limits and identifying discontinuities.

Example 4:

Consider the function: k(x) = (sin x) / x

This function has a removable discontinuity at x = 0 because it is undefined at x = 0. However, lim (x→0) (sin x) / x = 1 using L'Hôpital's rule or known trigonometric limits.

Practical Applications and Significance

The identification of removable discontinuities holds significant practical implications across various fields:

• Signal Processing: In analyzing signals, removable discontinuities can represent momentary interruptions or glitches that need to be addressed for accurate interpretation.

• Engineering: In design and modeling, identifying and handling removable discontinuities is crucial for ensuring the stability and reliability of systems.

• Physics: In physical models, discontinuities often indicate abrupt changes or transitions in physical quantities. Understanding their nature is important for accurate predictions.

• Computer Graphics: In generating smooth curves and surfaces, removable discontinuities can cause visual artifacts that need to be eliminated.

Beyond Identification: "Repairing" Removable Discontinuities

Once a removable discontinuity has been identified, it's often possible to "repair" the function by redefining its value at the point of discontinuity. This involves assigning the limit value to the function at the point of discontinuity, thereby creating a continuous extension of the function.

Example (revisiting Example 1):

The function f(x) = (x² - 4) / (x - 2) has a removable discontinuity at x = 2. The limit as x approaches 2 is 4. We can redefine the function as:

f*(x) = { (x² - 4) / (x - 2), x ≠ 2 { 4, x = 2

This redefined function f*(x) is now continuous at x = 2.

Conclusion

Identifying functions with removable discontinuities is a critical skill in calculus and related fields. By systematically analyzing the function's expression, calculating limits, and employing graphical interpretation, we can effectively pinpoint and characterize these discontinuities. Understanding removable discontinuities not only enhances mathematical comprehension but also provides practical tools for handling discontinuities in various real-world applications, paving the way for smoother, more robust models and systems. Remember to always combine algebraic analysis with graphical representation for a complete and accurate understanding. This approach ensures a thorough examination of the function's behavior and leads to confident identification of removable discontinuities.