Problem 61 - Brief Applied Calculus

Question

61-64. 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. $$ f(x)=\left\\{\begin{array}{ll} x & \text { if } x \leq 3 \\ 6-x & \text { if } x>3 \end{array}\right. $$

Step-by-Step Solution

Answer

The function is continuous at \( x = 3 \); all conditions of continuity are satisfied.

Step 1: Identify the Function Segments

We have a piecewise linear function defined as follows: For \( x \leq 3 \), the function is \( f(x) = x \). For \( x > 3 \), the function is \( f(x) = 6 - x \). These represent two linear equations that will be graphed based on these inequalities.

Step 2: Graph the Function

To graph the piecewise function, we'll consider each segment separately:1. **For** \( x \leq 3 \): Plot the line \( f(x) = x \). This is a diagonal line with a slope of 1. The graph should include all points from negative infinity up to \( x = 3 \), inclusive.2. **For** \( x > 3 \): Plot the line \( f(x) = 6 - x \). This line starts at \( x = 3 \) (but is not plotted at \( x = 3 \)) and goes to positive infinity, with a slope of -1.Note that there is a filled dot on the line \( y = x \) at \( x = 3 \), indicating that the function includes this point, and an open dot on the line \( y = 6 - x \) at \( x = 3 \).

Step 3: Find the Limit as x Approaches 3 from the Left

The left-hand limit as \( x \to 3^- \) is determined by the segment \( x \leq 3 \), which is \( f(x) = x \). Thus, \[\lim_{{x \to 3^-}} f(x) = 3\]

Step 4: Find the Limit as x Approaches 3 from the Right

The right-hand limit as \( x \to 3^+ \) is determined by the segment \( x > 3 \), which is \( f(x) = 6 - x \). Thus, \[\lim_{{x \to 3^+}} f(x) = 6 - 3 = 3\]

Step 5: Check Continuity at x = 3

To check for continuity at \( x = 3 \), ensure three conditions are satisfied:1. \( f(3) \) exists. - From \( f(x) = x \) when \( x = 3 \), we have \( f(3) = 3 \).2. \( \lim_{{x \to 3}} f(x) \) exists. - Both left and right limits equal 3.3. \( \lim_{{x \to 3}} f(x) = f(3) \).Since all three conditions are satisfied, the function is continuous at \( x = 3 \).

Key Concepts

Continuity

Understanding continuity in the context of piecewise functions is pivotal. Continuity at a point means that there are no jumps, holes, or interruptions at that point. For a function to be continuous at a specific point, three conditions must be fulfilled:

• The function must be defined at the point (i.e., it has a real value, not undefined).
• The limit of the function as it approaches the point from both directions must exist.
• The value of the function at that point must be equal to the limit of the function as it approaches the point.

In the given exercise, the function is continuous at \( x = 3 \) because all these conditions are met. This is crucial when graphing or analyzing the function because it means we can smoothly draw the line through that point without lifting our pencil.

Limits

Limits help us understand the behavior of functions as they approach specific points, which is especially useful in piecewise functions. The limit of a function at a point gives us information about where the function is heading as the input gets closer and closer to that point. To check a limit:

• Determine the value the function approaches from both sides.
• If both sides approach the same value, then the general limit exists at that point.

For our function, limits are considered at \( x = 3 \). Both the left-hand and right-hand limits equal \( 3 \), indicating that the function approaches the same value from either side, which is a peaceful transition, showcasing the behavior at that specific point.

Graphing Piecewise Functions

Graphing piecewise functions involves drawing different segments over specified intervals. These functions might not follow a single formula; instead, they're made up of several linear pieces, each applicable over different input ranges. The process involves:

• Identifying each piece or segment of the function.
• Plotting each segment over its respective interval.
• Ensuring that any endpoints are accurately represented (filled or open dots).

In the example provided, the function consists of two segments; a line with slope 1 for \( x \leq 3 \) and another line with slope -1 for \( x > 3 \). Properly graphing requires acknowledging where the segments connect and ensuring any discontinuity is marked appropriately.

Left-hand Limit

The left-hand limit describes the behavior of the function as the input approaches a specific point from the negative side. It's written as \( \lim_{{x \to c^-}} f(x) \), where \( c \) is the point of interest.

In our exercise, we evaluate the function as \( x \) approaches 3 from values less than 3. Given the rule \( f(x) = x \) for \( x \leq 3 \), the left-hand limit at \( x=3 \) is 3. This suggests the function is smoothly approaching a value of 3 from the left side, a crucial step for assessing continuity and determining the existence of a general limit at that point.

Right-hand Limit

The right-hand limit examines the behavior of the function as the input nears a particular point from the positive side. Represented as \( \lim_{{x \to c^+}} f(x) \), where \( c \) is the concerned point.

For the function in question, the right-hand limit as \( x \) approaches 3 must be examined for values greater than 3. With the defined segment \( f(x) = 6 - x \), the right-hand limit is also 3 as \( x \to 3^+ \). This consistency on both sides is key to continuity and ensures the function behaves predictably as it approaches our chosen value.