Analyzing G(x) Limits And Continuity At X=3
This article delves into the intricacies of a piecewise function, g(x), defined as follows:
g(x) = { 4x + 7, for 1 ≤ x < 3
5x + 3, for 3 ≤ x ≤ 5 }
We will embark on a comprehensive exploration of this function, meticulously examining its behavior as x approaches specific values, and ultimately determine its continuity at a crucial point. Our investigation will involve calculating function values at various points, evaluating limits from both sides, and applying the formal definition of continuity.
(a) Evaluating g(x) as x Approaches 3 from the Left
To understand the behavior of g(x) as x approaches 3 from the left, we need to evaluate the function at values progressively closer to 3 but less than 3. This involves using the first part of the piecewise function definition, g(x) = 4x + 7, which applies when 1 ≤ x < 3. We will calculate g(x) at x = 2.9, 2.99, and 2.999.
First, let's evaluate g(2.9). Substituting x = 2.9 into the expression 4x + 7, we get:
g(2.9) = 4(2.9) + 7 = 11.6 + 7 = 18.6
Next, we evaluate g(2.99) using the same formula:
g(2.99) = 4(2.99) + 7 = 11.96 + 7 = 18.96
Finally, we calculate g(2.999):
g(2.999) = 4(2.999) + 7 = 11.996 + 7 = 18.996
These calculations demonstrate a clear trend. As x gets closer and closer to 3 from the left side, the value of g(x) approaches 19. This observation is crucial for understanding the left-hand limit of the function at x = 3.
Analyzing these results, it's evident that the function values are increasing and converging towards a specific value as x approaches 3 from the left. This suggests the existence of a left-hand limit. The left-hand limit represents the value that a function approaches as the input variable approaches a given value from the left side. In this case, as x nears 3 from values less than 3, g(x) appears to be approaching 19. This convergence is a fundamental concept in calculus, allowing us to analyze function behavior near points where the function might not be directly defined or might exhibit unusual behavior.
Furthermore, understanding the left-hand limit is essential when dealing with piecewise functions. Piecewise functions are defined by different formulas over different intervals, and the limits at the boundaries of these intervals determine the function's overall behavior and continuity. The left-hand limit, in conjunction with the right-hand limit, provides a complete picture of how the function behaves around a particular point. By evaluating g(x) at values very close to 3 from the left, we gain valuable insight into the function's tendency and its potential for continuity at x = 3. This meticulous calculation forms the bedrock for a more thorough analysis of g(x)'s properties.
(b) Evaluating g(x) as x Approaches 3 from the Right
Now, let's investigate the behavior of g(x) as x approaches 3 from the right. This means we will evaluate the function at values progressively closer to 3 but greater than 3. For this, we use the second part of the piecewise function definition, g(x) = 5x + 3, which applies when 3 ≤ x ≤ 5. We will calculate g(x) at x = 3.1, 3.01, and 3.001.
First, let's evaluate g(3.1). Substituting x = 3.1 into the expression 5x + 3, we get:
g(3.1) = 5(3.1) + 3 = 15.5 + 3 = 18.5
Next, we evaluate g(3.01) using the same formula:
g(3.01) = 5(3.01) + 3 = 15.05 + 3 = 18.05
Finally, we calculate g(3.001):
g(3.001) = 5(3.001) + 3 = 15.005 + 3 = 18.005
These calculations, much like the left-hand limit analysis, reveal a distinct trend. As x gets increasingly closer to 3 from the right side, the value of g(x) approaches 18. This observation is essential for determining the right-hand limit of the function at x = 3. The contrast between this right-hand limit and the previously observed left-hand limit is a critical point in assessing the function's continuity at this point.
The right-hand limit is a fundamental concept in calculus that describes the value a function approaches as its input approaches a given value from the right side. In the context of our piecewise function g(x), the right-hand limit at x = 3 is determined by the portion of the function defined for values greater than or equal to 3, which is g(x) = 5x + 3. Our calculations at x = 3.1, 3.01, and 3.001 demonstrate a clear convergence towards 18 as x approaches 3 from the right. This limit is crucial for understanding the function's behavior as it transitions from one defined interval to another within its piecewise definition.
Understanding the right-hand limit is especially important when dealing with piecewise functions because it, along with the left-hand limit, helps determine whether the overall limit exists at the point where the function definition changes. If the right-hand limit and the left-hand limit exist and are equal at a given point, then the overall limit exists at that point. However, if these limits are different, as we are beginning to see in this case, the overall limit does not exist, which has significant implications for the function's continuity at that point. Therefore, a meticulous calculation and analysis of the right-hand limit are indispensable steps in fully characterizing the behavior of piecewise functions.
(c) Determining the Left-Hand and Right-Hand Limits at x = 3
From our previous calculations, we can now formally determine the left-hand and right-hand limits of g(x) as x approaches 3. The left-hand limit, denoted as lim (x→3-) g(x), is the value that g(x) approaches as x approaches 3 from values less than 3. As we saw in part (a), as x gets closer to 3 from the left, g(x) approaches 19. Therefore:
lim (x→3-) g(x) = 19
The right-hand limit, denoted as lim (x→3+) g(x), is the value that g(x) approaches as x approaches 3 from values greater than 3. From our calculations in part (b), we observed that as x gets closer to 3 from the right, g(x) approaches 18. Thus:
lim (x→3+) g(x) = 18
The fact that the left-hand limit (19) and the right-hand limit (18) are not equal has profound implications for the continuity of g(x) at x = 3. The existence and equality of one-sided limits are crucial for the existence of the overall limit, a fundamental requirement for continuity.
The distinct values of the left-hand and right-hand limits at x = 3 underscore a critical characteristic of the function g(x): it exhibits a jump discontinuity at this point. A jump discontinuity occurs when the one-sided limits exist but are not equal to each other. This means that as x approaches 3, the function
