Problem 55 - Calculus and its applications

Question

Is the function given by $$ G(x)=\left\\{\begin{array}{ll} \frac{1}{2} x+1, & \text { for } x<4 \\ -x+5, & \text { for } x>4 \end{array}\right. $$ continuous at \(x=4\) ? Why or why not?

Step-by-Step Solution

Answer

The function \(G(x)\) is not continuous at \(x=4\) because the left and right-hand limits differ.

Step 1: Definition of Continuity

A function is continuous at a point if the left-hand limit at that point equals the right-hand limit and both equal the function's value at that point. We must check these conditions for \(G(x)\) at \(x=4\).

Step 2: Calculate Left-Hand Limit at x=4

For \(x<4\), the function is given by \(\frac{1}{2}x + 1\). We calculate the left-hand limit as\[\lim_{{x \to 4^-}} G(x) = \lim_{{x \to 4^-}} \left(\frac{1}{2}x + 1\right).\]Substituting \(x = 4\), we get \[\lim_{{x \to 4^-}} G(x) = \frac{1}{2}(4) + 1 = 2 + 1 = 3.\]

Step 3: Calculate Right-Hand Limit at x=4

For \(x>4\), the function is given by \(-x + 5\). We calculate the right-hand limit as\[\lim_{{x \to 4^+}} G(x) = \lim_{{x \to 4^+}} (-x + 5).\]Substituting \(x = 4\), we get \[\lim_{{x \to 4^+}} G(x) = -4 + 5 = 1.\]

Step 4: Compare Limits and Function Value at x=4

Since \(x=4\) is not included in either domain piece, we do not have a function value at \(x=4\) in the piecewise definition.Contrast the left-hand limit, \(3\), and the right-hand limit, \(1\). They do not match, indicating that the limit of the function at \(x=4\) does not exist.

Step 5: Conclusion on Continuity

Since the left-hand and right-hand limits are not equal, \(G(x)\) is not continuous at \(x=4\).

Key Concepts

Piecewise Functions

Piecewise functions are unique in that their equations change based on the input value, or more specifically, the range of input values. This means that the function behaves differently in separate intervals of the domain, which can make analyzing them a bit challenging. Consider the function from our exercise:

• For values of "x" less than 4, the function is described by \( \frac{1}{2}x + 1 \).
• For "x" greater than 4, the function is given by \( -x + 5 \).

It's essential to note that piecewise functions can handle distinct behaviors at specific points, leading to interesting scenarios at transition points such as "x = 4". Here, each part of the function is defined separately, ensuring the function's unique behavior based on value ranges. When dealing with piecewise functions, always pay attention to how the segments of the function connect at the boundary points, as these determine the function's continuity.

Limits and Continuity

In calculus, understanding limits and continuity is crucial when dealing with piecewise functions. A function is continuous at a certain point if the following three conditions are satisfied:

• The function is defined at the specific point.
• The limit of the function exists at that point.
• The limit at that point equals the function's value.

For our given piecewise function, we examined its continuity at "x = 4." A notable hurdle arises because the function is not defined directly at this point. Thus, no single output covers "x = 4." Without a value at "x = 4", we rely on the limits from the left (as "x" approaches 4 from values less than 4) and the right (as "x" approaches 4 from values greater than 4) to decide continuity.
Continuous functions appear as unbroken lines on a graph; imagine a continuous path without lifting your pencil from paper. However, if any of the three conditions for continuity fails, as with our example function, it suggests a break or a gap, showing discontinuity at that point.

Evaluating Limits

Evaluating limits is a fundamental skill in calculus, especially when assessing a function's continuity. To determine the continuity at a specific point like "x = 4" in our piecewise function, we must compute the limits from both sides:

• **Left-Hand Limit**: Calculated for \( x<4 \), using \( \frac{1}{2}x + 1 \). As "x" approaches 4, this limit is 3.
• **Right-Hand Limit**: Evaluated for \( x>4 \), with \( -x + 5 \). As "x" approaches 4 again, this limit stands at 1.

These limits help us assess how the function behaves surrounding the point in question.
Since the left-hand and right-hand limits differ, this results in a non-existing overall limit at "x = 4." When these limits do not equal each other, it acts as a clear indicator that the function is not continuous at that point. Understanding how to calculate and compare these directional limits is crucial as it reveals function behavior near transition points, crucial for grasping broader calculus concepts.