Question 2: Exploring Limits And Continuity Of A Piecewise Function

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In the realm of calculus, piecewise functions present a unique challenge, requiring careful consideration of different definitions across various intervals. Question 2 delves into the intricacies of such a function, offering a multifaceted exploration of limits and continuity. This article aims to provide a comprehensive analysis of the function, its limits, and its continuity, offering insights and strategies for tackling similar problems.

Defining the Piecewise Function

At the heart of our exploration lies the piecewise function F(x), defined as follows:

F(x) = { sqrt(-x), if x < 0; 3-x, if 0 <= x < 3; (x-3)^2, if x > 3 }

This function exhibits distinct behaviors across different intervals. For negative values of x, it follows the square root of the negative of x. In the interval between 0 and 3 (inclusive of 0, exclusive of 3), it adheres to the linear function 3-x. Beyond 3, it transforms into a quadratic function, the square of (x-3). The absence of a definition at x = 3 is noteworthy, as it hints at potential discontinuities at this point. Understanding these nuances is paramount for accurately evaluating limits and assessing continuity.

(a) Evaluating the Limits

(i) Unveiling the Limit as x Approaches 0 from the Left: lim_(x→0-) F(x)

The first task is to evaluate the limit as x approaches 0 from the left, denoted as lim_(x→0-) F(x). When dealing with one-sided limits in piecewise functions, it is crucial to identify the relevant interval. As x approaches 0 from the left, we consider values of x that are less than 0. Therefore, we refer to the first piece of our function's definition: F(x) = √(-x) for x < 0.

Now, we can substitute x with values approaching 0 from the negative side. As x gets closer and closer to 0 from the left, -x approaches 0 from the positive side. Consequently, √(-x) approaches √0, which equals 0. Thus, we can confidently conclude that:

lim_(x→0-) F(x) = lim_(x→0-) √(-x) = 0

This limit signifies the function's behavior as it nears 0 from the left, providing a crucial piece of the puzzle in understanding the function's overall behavior around x = 0. The meticulous application of the relevant definition for the specified interval is essential in evaluating such limits.

(ii) Determining the Limit as x Approaches 0 from the Right: lim_(x→0+) F(x)

Next, we tackle the limit as x approaches 0 from the right, symbolized as lim_(x→0+) F(x). This time, we are interested in values of x that are greater than 0 but still approaching 0. This corresponds to the second piece of our piecewise function definition: F(x) = 3-x for 0 ≤ x < 3.

As x approaches 0 from the right, we can substitute values infinitesimally greater than 0 into the expression 3-x. As x gets arbitrarily close to 0, 3-x approaches 3-0, which equals 3. Therefore, we can definitively state that:

lim_(x→0+) F(x) = lim_(x→0+) (3-x) = 3

This limit reveals the function's trend as it approaches 0 from the right, providing a contrasting perspective to the left-hand limit. The discrepancy between the left-hand and right-hand limits at x = 0 hints at a potential discontinuity at this point.

(iii) Evaluating the Overall Limit as x Approaches 0: lim_(x→0) F(x)

Having computed the left-hand limit and the right-hand limit as x approaches 0, we now address the existence of the overall limit, lim_(x→0) F(x). A fundamental principle of calculus dictates that for a two-sided limit to exist at a point, the left-hand limit and the right-hand limit must both exist and be equal. In mathematical terms:

lim_(x→a) F(x) exists if and only if lim_(x→a-) F(x) = lim_(x→a+) F(x)

In our case, we have established that lim_(x→0-) F(x) = 0 and lim_(x→0+) F(x) = 3. Since 0 ≠ 3, the left-hand limit and the right-hand limit are not equal. Consequently, we conclude that the overall limit as x approaches 0 does not exist:

lim_(x→0) F(x) does not exist

This non-existence of the limit signifies a discontinuity at x = 0, a crucial characteristic of the function's behavior. Understanding the relationship between one-sided limits and the overall limit is paramount in analyzing continuity.

(iv) Probing the Limit as x Approaches 3: lim_(x→3) F(x)

Now, we shift our focus to the limit as x approaches 3, lim_(x→3) F(x). Given the piecewise nature of F(x) and the absence of a defined value at x = 3, we must again consider the left-hand and right-hand limits separately.

As x approaches 3 from the left (x→3-), we use the second piece of the function's definition: F(x) = 3-x for 0 ≤ x < 3. Thus, lim_(x→3-) F(x) = lim_(x→3-) (3-x) = 3-3 = 0.

As x approaches 3 from the right (x→3+), we employ the third piece of the function's definition: F(x) = (x-3)^2 for x > 3. Therefore, lim_(x→3+) F(x) = lim_(x→3+) (x-3)^2 = (3-3)^2 = 0.

Since the left-hand limit and the right-hand limit both exist and are equal to 0, we can conclude that the overall limit as x approaches 3 exists and is equal to 0:

lim_(x→3) F(x) = 0

This finding reveals that, despite the lack of a defined value at x = 3, the function approaches a specific value as x gets arbitrarily close to 3. This behavior is characteristic of a removable discontinuity, a concept we will explore further in the context of continuity.

(b) Unveiling the Nature of Continuity

The Essence of Continuity

The concept of continuity is central to the study of functions in calculus. Intuitively, a function is continuous at a point if its graph can be drawn without lifting the pen. More formally, a function F(x) is continuous at a point x = a if the following three conditions are met:

  1. F(a) is defined (i.e., the function has a value at x = a).
  2. lim_(x→a) F(x) exists (i.e., the limit of the function as x approaches a exists).
  3. lim_(x→a) F(x) = F(a) (i.e., the limit of the function as x approaches a is equal to the function's value at a).

If any of these conditions are not satisfied, the function is said to be discontinuous at x = a.

Examining Continuity at x = 0

Having evaluated the limits at x = 0, we are now equipped to assess the continuity of F(x) at this point. Recall the three conditions for continuity:

  1. F(0) is defined: According to the function's definition, F(0) = 3 - 0 = 3. Thus, the first condition is satisfied.
  2. lim_(x→0) F(x) exists: We previously established that lim_(x→0) F(x) does not exist. Therefore, the second condition is not met.
  3. lim_(x→0) F(x) = F(0): Since the limit does not exist, this condition cannot be satisfied.

Since the second and third conditions for continuity are not satisfied at x = 0, we conclude that F(x) is discontinuous at x = 0. The non-existence of the limit signifies a jump discontinuity, where the function abruptly jumps from one value to another as x passes through 0. This jump discontinuity is visually evident in the graph of the function.

Investigating Continuity at x = 3

Next, we turn our attention to the continuity of F(x) at x = 3. Once again, we consider the three conditions for continuity:

  1. F(3) is defined: Examining the function's definition, we observe that F(3) is not defined. There is no piece of the function that applies specifically when x = 3. Thus, the first condition is not satisfied.
  2. lim_(x→3) F(x) exists: We previously determined that lim_(x→3) F(x) = 0. Therefore, the second condition is satisfied.
  3. lim_(x→3) F(x) = F(3): Since F(3) is not defined, this condition cannot be satisfied.

As the first and third conditions for continuity are not met at x = 3, we conclude that F(x) is discontinuous at x = 3. This discontinuity is classified as a removable discontinuity. A removable discontinuity occurs when the limit of the function exists at a point, but the function is either undefined at that point or the function's value at that point does not match the limit. In this case, if we were to redefine F(3) as 0, the function would become continuous at x = 3, effectively "removing" the discontinuity. Understanding the different types of discontinuities is essential for a complete analysis of a function's behavior.

Conclusion

Through a meticulous evaluation of limits and an application of the continuity criteria, we have gained a comprehensive understanding of the piecewise function F(x). We identified a jump discontinuity at x = 0, stemming from the differing left-hand and right-hand limits. At x = 3, we uncovered a removable discontinuity, characterized by the existence of a limit despite the function being undefined at that point. This analysis highlights the importance of considering one-sided limits and the formal definition of continuity when dealing with piecewise functions. By mastering these concepts, one can confidently navigate the complexities of calculus and gain deeper insights into the behavior of functions.