Limit Existence Of Piecewise Function F(x) At X=3

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Introduction

In the realm of calculus, the concept of limits forms the bedrock upon which continuity, derivatives, and integrals are built. Understanding limits is crucial for grasping the behavior of functions as their input approaches a specific value. This article delves into a fascinating case: the function

f(x) = \begin{cases} \frac{|x-3|}{x-3}, & x \neq 3 \\ 1, & x = 3. \end{cases}

Our primary objective is to rigorously determine whether the limit of f(x) exists as x approaches 3. This exploration will involve a meticulous examination of the function's behavior from both the left and the right of x = 3, shedding light on the fundamental principles governing the existence of limits. The existence of a limit at a point is a cornerstone concept in calculus, dictating whether a function smoothly approaches a particular value as its input gets arbitrarily close to that point. In the given scenario, we're presented with a piecewise function, which introduces an added layer of complexity. Piecewise functions are defined by different expressions over different intervals, making it imperative to analyze each piece separately when evaluating limits. The absolute value within the function further necessitates a careful consideration of cases, as the behavior of |x - 3| differs depending on whether x is greater or less than 3. By systematically dissecting the function and employing the formal definition of limits, we aim to provide a comprehensive answer to the question at hand.

Understanding the Function

To effectively analyze the limit, we must first dissect the function f(x). This function is defined piecewise, meaning its behavior is described by different formulas over different intervals. For x not equal to 3, the function is defined as f(x) = |x - 3| / (x - 3). This expression involves an absolute value, which necessitates considering two distinct cases:

  1. When x > 3, x - 3 is positive, so |x - 3| = x - 3. Thus, f(x) = (x - 3) / (x - 3) = 1.
  2. When x < 3, x - 3 is negative, so |x - 3| = -(x - 3). Thus, f(x) = -(x - 3) / (x - 3) = -1.

At the specific point x = 3, the function is defined as f(3) = 1. This direct assignment of a value at x = 3 is crucial to note, as it may or may not align with the function's limiting behavior as x approaches 3. The interplay between the function's piecewise definition and the concept of limits is central to this analysis. Specifically, the existence of a limit as x approaches 3 hinges on whether the function values converge to the same value from both the left and the right. If the left-hand limit and the right-hand limit exist and are equal, then the limit exists. However, if these one-sided limits differ, the overall limit does not exist. In the context of our function, this means we need to carefully examine how f(x) behaves as x gets closer to 3 from values greater than 3 and from values less than 3. By understanding the function's behavior in these two distinct scenarios, we can then compare the resulting limits and draw a definitive conclusion about the existence of the limit at x = 3.

Evaluating the Left-Hand Limit

The left-hand limit examines the behavior of f(x) as x approaches 3 from values less than 3. Mathematically, this is denoted as lim ₓ→₃⁻ f(x). As established earlier, when x < 3, f(x) = -1. This means that regardless of how close x gets to 3 from the left, the function's value remains constant at -1. Therefore, the left-hand limit is:

lim ₓ→₃⁻ f(x) = lim ₓ→₃⁻ -1 = -1

This result is a direct consequence of the piecewise definition of the function. For all x less than 3, f(x) is explicitly defined as -1. Thus, as x approaches 3 from the left, there is no change in the function's value; it remains steadfastly at -1. This constant behavior greatly simplifies the evaluation of the left-hand limit. It eliminates any ambiguity or need for more complex limit-finding techniques. The left-hand limit serves as a crucial piece of the puzzle when determining the overall limit's existence. The other critical piece is the right-hand limit, which examines the function's behavior as x approaches 3 from values greater than 3. To assert that the limit exists at x = 3, the left-hand limit and the right-hand limit must not only exist individually but also be equal to each other. If they differ, then the overall limit is deemed non-existent. This requirement highlights the importance of evaluating both one-sided limits in our analysis. In the next section, we will turn our attention to evaluating the right-hand limit of f(x) as x approaches 3.

Evaluating the Right-Hand Limit

Now, let's investigate the right-hand limit, which concerns the behavior of f(x) as x approaches 3 from values greater than 3. This is written as lim ₓ→₃⁺ f(x). From our earlier analysis, we know that when x > 3, f(x) = 1. Consequently, as x approaches 3 from the right, the function's value remains constant at 1. Therefore, the right-hand limit is:

lim ₓ→₃⁺ f(x) = lim ₓ→₃⁺ 1 = 1

Similar to the left-hand limit, the right-hand limit's value is a direct result of the function's piecewise definition. For all x greater than 3, f(x) is explicitly defined as 1. This means that as x gets arbitrarily close to 3 from the right, the function's value does not fluctuate; it remains firmly at 1. This consistent behavior simplifies the process of finding the right-hand limit. The right-hand limit, alongside the previously calculated left-hand limit, is a pivotal component in determining whether the overall limit of f(x) exists at x = 3. As mentioned earlier, for the limit to exist, the left-hand limit and the right-hand limit must converge to the same value. In our case, we've found that the left-hand limit is -1, and the right-hand limit is 1. The discrepancy between these two values leads us to a significant conclusion about the limit's existence, which we will discuss in detail in the subsequent section. The comparison of these one-sided limits is a fundamental technique in calculus for assessing the existence of limits, particularly for piecewise functions where the function's behavior may differ on either side of a specific point.

Determining the Existence of the Limit

To definitively determine if the limit of f(x) exists as x approaches 3, we must compare the left-hand limit and the right-hand limit. We have established that:

  • The left-hand limit, lim ₓ→₃⁻ f(x) = -1
  • The right-hand limit, lim ₓ→₃⁺ f(x) = 1

The fundamental principle of limits dictates that for a limit to exist at a point, the left-hand limit and the right-hand limit must both exist and be equal. In our case, the left-hand limit is -1, and the right-hand limit is 1. Since -1 ≠ 1, the left-hand limit and the right-hand limit are not equal. Therefore, we can conclude that the limit of f(x) as x approaches 3 does not exist. This non-existence of the limit has significant implications for the function's behavior at and around x = 3. It indicates a discontinuity in the function at this point. A discontinuity occurs when a function does not have a smooth, unbroken graph at a particular point. In this specific case, the function jumps from -1 to 1 as x crosses 3, creating a break in the graph. The concept of limits is intimately connected to continuity. If a function has a limit at a point and the limit's value matches the function's value at that point, then the function is continuous at that point. However, in our situation, the limit does not exist, so the function cannot be continuous at x = 3. The discontinuity we observe here is a jump discontinuity, a specific type of discontinuity where the function abruptly changes value. This kind of analysis is crucial in calculus for understanding the nature and behavior of functions, particularly in identifying points where a function may exhibit unusual or non-smooth characteristics.

Conclusion

In conclusion, after a thorough analysis of the function f(x) = egin{cases} \frac{|x-3|}{x-3}, & x \neq 3 \ 1, & x = 3. \end{cases}, we have determined that the limit as x approaches 3 does not exist. This conclusion is supported by the fact that the left-hand limit (-1) and the right-hand limit (1) are not equal. This exploration highlights the critical role of one-sided limits in determining the existence of a two-sided limit. The concept of limits is fundamental to calculus, serving as the foundation for understanding continuity, derivatives, and integrals. When the left-hand and right-hand limits disagree, as in this case, it signifies a discontinuity in the function's graph at the point in question. This jump discontinuity at x = 3 is a key characteristic of the function f(x). The insights gained from this analysis have broader implications in the study of functions and their behavior. The ability to rigorously evaluate limits and identify discontinuities is essential for understanding the properties of various functions and their applications in diverse fields. This process of evaluating limits, especially for piecewise functions, underscores the importance of careful analysis and a solid grasp of the underlying principles of calculus. By systematically dissecting the function's behavior from both sides of the point of interest, we can arrive at definitive conclusions about the existence and value of limits, furthering our understanding of the function's overall characteristics.