Problem 62 - Brief Applied Calculus

Question

For each piecewise linear function: a. Draw its graph (by hand or using a graphing calculator). b. Find the limits as \(x\) approaches 3 from the left and from the right. c. Is it continuous at \(x=3\) ? If not, indicate the first of the three conditions in the definition of continuity (page 87) that is violated. $$ f(x)=\left\\{\begin{array}{ll} 5-x & \text { if } x \leq 3 \\ x-2 & \text { if } x>3 \end{array}\right. $$

Step-by-Step Solution

Answer

Not continuous at \(x=3\) as left limit \(\neq\) right limit.

Step 1: Graphing the Function

To draw the graph of the piecewise function, analyze each linear piece separately. For the first piece, when \(x \leq 3\), the function is given by \(f(x) = 5-x\). This is a line with a slope of \(-1\) and a y-intercept at \(y=5\). This part of the graph includes the point at \(x=3\), resulting in a filled circle at \((3, 2)\). For the second piece, when \(x > 3\), the function is \(f(x) = x - 2\), which is a line with a slope of 1 and a y-intercept at \(y = -2\). This part starts at \(x=3\) but does not include this point, represented by an open circle at \((3, 1)\). Plot these two lines on the same set of axes.

Step 2: Finding the Left-Hand Limit as x Approaches 3

The left-hand limit is determined by the value of \(f(x)\) as \(x\) approaches 3 from values less than 3. Since \(x\leq3\), the function \(f(x) = 5-x\) is used. Thus, compute the limit: \(\lim_{{x \to 3^-}} f(x) = 5-3 = 2\).

Step 3: Finding the Right-Hand Limit as x Approaches 3

The right-hand limit is determined by the value of \(f(x)\) as \(x\) approaches 3 from values greater than 3. Here, the function \(f(x) = x-2\) applies. Compute the limit: \(\lim_{{x \to 3^+}} f(x) = 3-2 = 1\).

Step 4: Determining Continuity at x = 3

A function is continuous at a point \(x=c\) if the following three conditions are satisfied: 1) \(f(c)\) is defined, 2) \(\lim_{{x \to c^-}} f(x) = \lim_{{x \to c^+}} f(x)\), and 3) \(\lim_{{x \to c}} f(x) = f(c)\). For \(x=3\), \(f(3)=5-3=2\). However, since the left-hand limit is 2 and the right-hand limit is 1, condition 2 fails. Therefore, the function is not continuous at \(x=3\). The condition that is violated is that the left-hand limit does not equal the right-hand limit at \(x=3\).

Key Concepts

Continuity

Continuity is a crucial concept in piecewise linear functions and calculus in general. A function is said to be continuous at a certain point if there are no breaks, jumps, or holes at that point. In mathematical terms, for a function to be continuous at a point \(x=c\), three conditions must be met:
- The function \(f\) must be defined at \(x=c\), meaning \(f(c)\) exists.- The limit of the function as \(x\) approaches \(c\) from the left must equal the limit from the right: \(\lim_{{x \to c^-}} f(x) = \lim_{{x \to c^+}} f(x)\).- The limit of the function as \(x\) approaches \(c\) must equal the actual function value at \(c\): \(\lim_{{x \to c}} f(x) = f(c)\).
In our example of the piecewise linear function \(f(x)\), it is not continuous at \(x=3\). Although \(f(3)\) exists and equals 2, the second condition fails as the left-hand limit (2) does not match the right-hand limit (1). This indicates a jump discontinuity at \(x=3\), which is common in piecewise functions designed to switch formulas at particular points.
Understanding continuity helps identify where and why a function might have breaks, which is essential for proper function analysis and graphing.

Limits

Limits describe the behavior of a function as the input approaches a certain value. In the context of piecewise linear functions, limits can reveal crucial information about the function's continuity.
To find the left-hand limit as \(x\) approaches a point from the left, we consider the values of \(x\) just less than the point. Conversely, the right-hand limit is found by considering values just greater than the point. The function in question, \(f(x)\), has different formulas based on whether \(x\) is less than or greater than 3. Therefore, we use different limits:

• Left-hand limit: As \(x\) approaches 3 from the left, we take \(f(x) = 5-x\). Calculating, \(\lim_{{x \to 3^-}} f(x) = 2\).
• Right-hand limit: As \(x\) approaches 3 from the right, \(f(x) = x-2\). Calculating, \(\lim_{{x \to 3^+}} f(x) = 1\).

The discrepancy between the left-hand and right-hand limits at \(x=3\) indicates a break. These calculations help pinpoint where a function might not be continuous by showing that \(\lim_{{x \to c^-}} f(x) eq \lim_{{x \to c^+}} f(x)\). Understanding limits is crucial for identifying potential changes in function behavior and for examining how the function behaves around specific points.

Graphical Representation

Graphical representation of functions provides a visual understanding that can sometimes be more intuitive than numerical analysis. For piecewise linear functions, graphing the function can quickly reveal its behavior and continuity.
When graphing, each piece of the function is plotted separately. In this case, \(f(x) = 5-x\) for \(x \leq 3\) and \(f(x) = x-2\) for \(x > 3\). The first piece plots as a line with a slope of \(-1\) which decreases as \(x\) increases, and includes the point \((3, 2)\) with a filled circle. The second piece is a line with a positive slope that doesn't include the starting point \((3, 1)\), shown by an open circle.

• Filled circle at \((3, 2)\): Represents inclusion, indicating that the point \((3, 2)\) is part of the function for \(x \leq 3\).
• Open circle at \((3, 1)\): Indicates exclusion, showing that while the line starts there, the point itself is not part of the function for \(x > 3\).

By plotting these segments on the same graph, the jump at \(x=3\) becomes visual. This graphical insight confirms that \(f(x)\) is not continuous at \(x=3\) due to the gap between the filled and open circles. Graphs serve as excellent tools to complement the numerical analysis, providing clarity by showing the exact nature of any discontinuity directly.